Definite Integral

3.3 Definite Integral
 
The image shows a mathematical concept related to the value of a definite integral. At the top, there is the text 'VALUE OF DEFINITE INTEGRAL' surrounded by decorative stars. Below this, there is a mathematical formula enclosed in a blue box. The formula represents the definite integral of a function f(x) from a to b, and it shows the steps to evaluate it using the antiderivative g(x). The formula is: ∫[a to b] f(x) dx = g(x) + C |[a to b] = [g(b) + C] - [g(a) + C] = g(b) - g(a) At the bottom of the box, there is the logo of Pandai.
 
Characteristics of Definite Integrals

For the function \(f(x)\) and \(g(x)\),

  • \(\int_{a}^{b} f(x) \ dx = 0\)
  • \(\int_{a}^{b} f(x) \ dx = -\int_{a}^{b} f(x) \ dx\)
  • \(\int_{a}^{b} kf(x) \ dx = k\int_{a}^{b} f(x) \ dx\), where \(k\) is a constant
  • \(\int_{a}^{b} f(x) \ dx + \int_{b}^{c} f(x) \ dx = \int_{a}^{c} f(x) \ dx\), where \(a \lt b \lt c\)
  • \(\int_{a}^{b} [f(x) \pm g(x)] \ dx = \int_{a}^{b} f(x) \ dx \pm \int_{a}^{b}g(x) \ dx\)
 
Area of Region between the Curve and the \(x\)-axis
Diagram

A graph depicting the function y=f(x) with a highlighted point and the area A shaded beneath the curve.

Formula

The formula for the area of the region \(A\) is given by:

\(A=\int^b_ay \, dx\)

 
Area of the Above and Below the \(x\)-axis
Figure

A graph depicting a function y=f(x), showing areas above and below the x-axis.

Description

For the value of the area bounded by the curve and the \(x\)-axis,

  • If the region is below the \(x\)-axis, then the integral value is negative.
  • If the region is above the \(x\)-axis, the the integral value is positive.
  • The areas of both regions are positive.
 
Area between the Curve and the \(y\)-axis
Figure

A graph depicting the function x=g(y) with a highlighted point and the area of region A shaded for emphasis.

Formula

The formula for the area of the region \(A\) is given by:

\(A=\int_a^b x \, dy\)

 
Area at the Right and Left of the \(y\)-aixs
Figure

An equation illustrating the integral value of a function represented as x = f(y), showcasing mathematical relationships.

Description

For a region bounded by the curve and the \(y\)-axis,

  • If the region is to the left of \(y\)-axis, then the integral value is negative.
  • If the region is to the right of \(y\)-axis, then the integral value is positive.
  • The areas of both regions are positive.
 
Area between the Curve and a Straight Line
Figure

A graph depicting a line and a curve, intersecting at points x=a and x=b, illustrating the functions y=f(x) and y=g(x).

Formula

The area of the shaded region can be illustrated as follows:

\(\begin{aligned} \text{The area of shaded region }&= \int_{a}^{b} g(x) \, dx -\int_{a}^{b} f(x) \, dx \\\\ &=\int_{a}^{b} [g(x)-f(x)] \, dx \end{aligned}\)

To find the area, make sure that the function of the curve or line on the top is substracted to the function of the curve or line at the bottom.

 
The Generated Volume of a Region Revolved at the Axis
Description
  • The generated volume of a solid when an area is rotated through \(360^\circ\) about the \(x\)-axis or \(y\)-axis.
  • The value of generated volume is always positive.
The Generated Volume at the \(x\)-axis

The generated volume \(V\) when a region bounded by the curve \(y=f(x)\) enclosed by \(x=a\) and \(x=b\) is revolved through \(360^\circ\) about the \(x\)-axis is given by:

\(V=\int_a^b \pi y^2 \, dx\)

The Generated Volume at the \(y\)-axis

The generated volume \(V\) when a region bounded by the curve \(x=g(y)\) enclosed by \(y=a\) and \(y=b\) is revolved through \(360^\circ\) about the \(y\)-axis is given by:

\(V=\int_a^b \pi x^2 \, dy\)

 
Example \(1\)
Question

A graph displaying a line and a curve, highlighting the shaded region between y=x^2 and y=x^{1/3} at points (0,0) and (1,1).

The curves \(y=x^2\) and \(y= \sqrt[3] {x}\) intersect at points \((0,0)\) and \((1,1)\). Find the area between the two curves.

Solution

\(\begin{aligned} \text{Area of the region }&= \int_{0}^{1} \sqrt[3] x \ dx - \int_{0}^{1} x^2 \ dx\\\\ &= \int_{0}^{1} (x^{\frac{1}{3}}-x^2) \ dx\\\\ &=\begin{bmatrix} \dfrac{3x^{\frac{4}{3}}}{4}-\dfrac{x^3}{3} \end{bmatrix}_{0}^{1}\\\\ &=\begin{bmatrix} \dfrac{3(1)^{\frac{4}{3}}}{4}-\dfrac{1^3}{3} \end{bmatrix}_{0}^{1} - \begin{bmatrix} \dfrac{3(0)^{\frac{4}{3}}}{4}-\dfrac{0^3}{3} \end{bmatrix}_{0}^{1}\\\\ &=\dfrac{5}{12} \text{ unit}^2 .\end{aligned}\)

 
Example \(2\)
Question

A graph illustrating a line and a curve, with a shaded area between their intersection points at (0,0) and (4,4).

Based on the diagram above, the curve \(y=\dfrac{1}{4}x^2\) intersects the stright line  \(y=x\) at \(O(0,0)\) and \(A(4,4)\). Find the generated volume, in terms of \(\pi\), when the shaded region is revolved fully about the \(x\)-axis.

Solution

Let \(V_{1} \) the volume generated by the straight line \(y=x\) and \(V_{2} \) be the volume generated by the curve  \(y=\dfrac{1}{4}x^2\) from \(x=0\) to \(x=4\).


\(\begin{aligned} V_1 &= \int_{0}^{4} \pi (x)^2 \ dx\\\\ &=\pi \int_{0}^{4} (x)^2 \ dx\\\\ &= \pi \begin{bmatrix} \dfrac{x^3}{3} \end{bmatrix}_{0}^{4}\\\\ &= \pi \begin{bmatrix} \dfrac{4^3}{3} - \dfrac{0^3}{3} \end{bmatrix} \\\\ &=\dfrac{64}{3}\pi \text{ unit}^3. \end{aligned}\)


\(\begin{aligned} V_2 &= \int_{0}^{4} \pi \begin{pmatrix} \dfrac{1}{4}x^2 \end{pmatrix}^2 \ dx\\\\ &=\pi \int_{0}^{4} \dfrac{1}{16}x^4 \ dx\\\\ &= \pi \begin{bmatrix} \dfrac{x^5}{16(5)} \end{bmatrix}_{0}^{4}\\\\ &= \pi \begin{bmatrix} \dfrac{4^5}{80} - \dfrac{0^5}{80} \end{bmatrix} \\\\ &=\dfrac{64}{5}\pi \text{ unit}^3. \end{aligned}\)


Thus, the generated volume

\(\begin{aligned} &= V_1 - V_2\\\\ &=\dfrac{64}{3} \pi -\dfrac{64}{5} \pi\\\\ &=8\dfrac{8}{15}\pi \text{ unit}^3. \end{aligned}\)

 

Definite Integral

3.3 Definite Integral
 
The image shows a mathematical concept related to the value of a definite integral. At the top, there is the text 'VALUE OF DEFINITE INTEGRAL' surrounded by decorative stars. Below this, there is a mathematical formula enclosed in a blue box. The formula represents the definite integral of a function f(x) from a to b, and it shows the steps to evaluate it using the antiderivative g(x). The formula is: ∫[a to b] f(x) dx = g(x) + C |[a to b] = [g(b) + C] - [g(a) + C] = g(b) - g(a) At the bottom of the box, there is the logo of Pandai.
 
Characteristics of Definite Integrals

For the function \(f(x)\) and \(g(x)\),

  • \(\int_{a}^{b} f(x) \ dx = 0\)
  • \(\int_{a}^{b} f(x) \ dx = -\int_{a}^{b} f(x) \ dx\)
  • \(\int_{a}^{b} kf(x) \ dx = k\int_{a}^{b} f(x) \ dx\), where \(k\) is a constant
  • \(\int_{a}^{b} f(x) \ dx + \int_{b}^{c} f(x) \ dx = \int_{a}^{c} f(x) \ dx\), where \(a \lt b \lt c\)
  • \(\int_{a}^{b} [f(x) \pm g(x)] \ dx = \int_{a}^{b} f(x) \ dx \pm \int_{a}^{b}g(x) \ dx\)
 
Area of Region between the Curve and the \(x\)-axis
Diagram

A graph depicting the function y=f(x) with a highlighted point and the area A shaded beneath the curve.

Formula

The formula for the area of the region \(A\) is given by:

\(A=\int^b_ay \, dx\)

 
Area of the Above and Below the \(x\)-axis
Figure

A graph depicting a function y=f(x), showing areas above and below the x-axis.

Description

For the value of the area bounded by the curve and the \(x\)-axis,

  • If the region is below the \(x\)-axis, then the integral value is negative.
  • If the region is above the \(x\)-axis, the the integral value is positive.
  • The areas of both regions are positive.
 
Area between the Curve and the \(y\)-axis
Figure

A graph depicting the function x=g(y) with a highlighted point and the area of region A shaded for emphasis.

Formula

The formula for the area of the region \(A\) is given by:

\(A=\int_a^b x \, dy\)

 
Area at the Right and Left of the \(y\)-aixs
Figure

An equation illustrating the integral value of a function represented as x = f(y), showcasing mathematical relationships.

Description

For a region bounded by the curve and the \(y\)-axis,

  • If the region is to the left of \(y\)-axis, then the integral value is negative.
  • If the region is to the right of \(y\)-axis, then the integral value is positive.
  • The areas of both regions are positive.
 
Area between the Curve and a Straight Line
Figure

A graph depicting a line and a curve, intersecting at points x=a and x=b, illustrating the functions y=f(x) and y=g(x).

Formula

The area of the shaded region can be illustrated as follows:

\(\begin{aligned} \text{The area of shaded region }&= \int_{a}^{b} g(x) \, dx -\int_{a}^{b} f(x) \, dx \\\\ &=\int_{a}^{b} [g(x)-f(x)] \, dx \end{aligned}\)

To find the area, make sure that the function of the curve or line on the top is substracted to the function of the curve or line at the bottom.

 
The Generated Volume of a Region Revolved at the Axis
Description
  • The generated volume of a solid when an area is rotated through \(360^\circ\) about the \(x\)-axis or \(y\)-axis.
  • The value of generated volume is always positive.
The Generated Volume at the \(x\)-axis

The generated volume \(V\) when a region bounded by the curve \(y=f(x)\) enclosed by \(x=a\) and \(x=b\) is revolved through \(360^\circ\) about the \(x\)-axis is given by:

\(V=\int_a^b \pi y^2 \, dx\)

The Generated Volume at the \(y\)-axis

The generated volume \(V\) when a region bounded by the curve \(x=g(y)\) enclosed by \(y=a\) and \(y=b\) is revolved through \(360^\circ\) about the \(y\)-axis is given by:

\(V=\int_a^b \pi x^2 \, dy\)

 
Example \(1\)
Question

A graph displaying a line and a curve, highlighting the shaded region between y=x^2 and y=x^{1/3} at points (0,0) and (1,1).

The curves \(y=x^2\) and \(y= \sqrt[3] {x}\) intersect at points \((0,0)\) and \((1,1)\). Find the area between the two curves.

Solution

\(\begin{aligned} \text{Area of the region }&= \int_{0}^{1} \sqrt[3] x \ dx - \int_{0}^{1} x^2 \ dx\\\\ &= \int_{0}^{1} (x^{\frac{1}{3}}-x^2) \ dx\\\\ &=\begin{bmatrix} \dfrac{3x^{\frac{4}{3}}}{4}-\dfrac{x^3}{3} \end{bmatrix}_{0}^{1}\\\\ &=\begin{bmatrix} \dfrac{3(1)^{\frac{4}{3}}}{4}-\dfrac{1^3}{3} \end{bmatrix}_{0}^{1} - \begin{bmatrix} \dfrac{3(0)^{\frac{4}{3}}}{4}-\dfrac{0^3}{3} \end{bmatrix}_{0}^{1}\\\\ &=\dfrac{5}{12} \text{ unit}^2 .\end{aligned}\)

 
Example \(2\)
Question

A graph illustrating a line and a curve, with a shaded area between their intersection points at (0,0) and (4,4).

Based on the diagram above, the curve \(y=\dfrac{1}{4}x^2\) intersects the stright line  \(y=x\) at \(O(0,0)\) and \(A(4,4)\). Find the generated volume, in terms of \(\pi\), when the shaded region is revolved fully about the \(x\)-axis.

Solution

Let \(V_{1} \) the volume generated by the straight line \(y=x\) and \(V_{2} \) be the volume generated by the curve  \(y=\dfrac{1}{4}x^2\) from \(x=0\) to \(x=4\).


\(\begin{aligned} V_1 &= \int_{0}^{4} \pi (x)^2 \ dx\\\\ &=\pi \int_{0}^{4} (x)^2 \ dx\\\\ &= \pi \begin{bmatrix} \dfrac{x^3}{3} \end{bmatrix}_{0}^{4}\\\\ &= \pi \begin{bmatrix} \dfrac{4^3}{3} - \dfrac{0^3}{3} \end{bmatrix} \\\\ &=\dfrac{64}{3}\pi \text{ unit}^3. \end{aligned}\)


\(\begin{aligned} V_2 &= \int_{0}^{4} \pi \begin{pmatrix} \dfrac{1}{4}x^2 \end{pmatrix}^2 \ dx\\\\ &=\pi \int_{0}^{4} \dfrac{1}{16}x^4 \ dx\\\\ &= \pi \begin{bmatrix} \dfrac{x^5}{16(5)} \end{bmatrix}_{0}^{4}\\\\ &= \pi \begin{bmatrix} \dfrac{4^5}{80} - \dfrac{0^5}{80} \end{bmatrix} \\\\ &=\dfrac{64}{5}\pi \text{ unit}^3. \end{aligned}\)


Thus, the generated volume

\(\begin{aligned} &= V_1 - V_2\\\\ &=\dfrac{64}{3} \pi -\dfrac{64}{5} \pi\\\\ &=8\dfrac{8}{15}\pi \text{ unit}^3. \end{aligned}\)