15C - Definite integrals
Definite integrals are integrals that contain limits/bounds of integration, for example: \[\int_{a}^{b} f(x)dx\] The statement above means that we integrate the function \(f(x)\) between \(x=a\) and \(x=b\). The values of \(a\) and \(b\) are known as the limits of integration.
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The fundamental theorem of calculus:
The fundamental theorem of calculus provides a way of evaluating definite integrals. If \(f(x)\) is a continuous function over the interval \(x\in [a,b]\), then
\[\int_{a}^{b} f(x) dx=[F(x)]_{a}^{b}=F(b)-F(a)\]
15C - Example 1: Evaluating definite integrals
Evaluate the following definite integral: \[\int_{-1}^{4} x^2 dx\]
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15C - Example 1: Video Example
15C - Example 1: Practice
Question 1:
Evaluate each of the following definite integrals.
15C - Example 1: Solutions
Question 1:
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15C - Example 2: Evaluating definite integrals
Evaluate the following definite integral: \[\int_{0}^{2} \frac{4}{5}x^3 dx\]
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15C - Example 2: Video Example
15C - Example 2: Practice
Question 1: Evaluate the following definite integral \[\int_{-2}^{-1}5x^4 dx\] Question 2: Find the value of \(a\) such that \[\int_{1}^{a}x^2dx=21\] 15C - Example 2: Solutions
Question 1: \[\int_{-2}^{-1}5x^4 dx = [x^4]_{-2}^{-1}\] \[=(-1)^5-(-2)^5=(-1)-(-32)=31\] Question 2: \[[\frac{x^3}{3}]_{1}^{a}=21\] \[\therefore (\frac{(a)^3}{3})-(\frac{(1)^3}{3})=21\] \[\therefore a^3-1=63\] \[\therefore a^3=64 \therefore a=4\] |
15C - Example 3: Evaluating definite integrals
Evaluate the following definite integral: \[\int_{1}^{4} 4\sqrt{x} dx\]
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15C - Example 3: Video Example
15C - Example 3: Practice
Question 1: Evaluate the following definite integral: \[\int_{1}^{5} \frac{1}{x^2} dx\] Question 2: Find the value of \(A\) such that \[\int_{1}^{9} \frac{A}{2\sqrt{x}}dx=6\] 15C - Example 3: Solutions
Question 1: \[\int_{1}^{5} x^{-2} dx=[-\frac{1}{x}]_{1}^{5}\] \[\therefore (-\frac{1}{5})-(-\frac{1}{1})=\frac{4}{5}\] Question 2: \[ [A\sqrt{x}]_{1}^{9}=6\] \[\therefore (A\sqrt{9})-(A\sqrt{1})=6\] \[\therefore 2A=6 \therefore A=3\] |
Definite integrals and their properties:
Definite integrals have limits/bounds of integration. \[\int_{a}^{b} f(x)dx\] When working with definite integrals, the following properties apply:
- For a function \(f(x)\) and a constant \(k\in R\): \[\int_{a}^{b} kf(x)dx=k \int_{a}^{b} f(x) dx\]
- For functions \(f(x)\) and \(g(x)\): \[\int_{a}^{b} f(x) \pm g(x) dx=\int_{a}^{b} f(x) dx \pm \int_{a}^{b} g(x) dx\]
- For a function \(f(x)\): \[\int_{a}^{b} f(x) dx = - \int_{b}^{a} f(x) dx\]
- For a function \(f(x)\) and constants \(a<c<b\): \[\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx\]
15C - Example 4: Evaluating definite integrals
Evaluate the following definite integral: \[\int_{-2}^{3} 6x^2-4x dx\]
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15C - Example 4: Video Example
15C - Example 4: Practice
Question 1: Evaluate the following definite integral \[\int_{-1}^{2}5-3x^2dx\] Question 2: Find the value of \(k\) such that \[\int_{0}^{4}k-\sqrt{x} dx=\frac{8}{3}\] 15C - Example 4: Solutions
Question 1: \[\int_{-1}^{2}5-3x^2dx=[5x-x^3]_{-1}^{2}\] \[=(5(2)-(2)^3)-(5(-1)-(-1)^3)\] \[=(10-8)-(-5+1)=6\] Question 2: \[ [kx-\frac{2x^{\frac{3}{2}}}{3}]_{0}^{4}=\frac{8}{3}\] \[\therefore (4k-\frac{16}{3})-(0)=\frac{8}{3}\] \[\therefore k=2\] |
15C - Example 5: Evaluating definite integrals (CAS)
Evaluate the following definite integral: \[\int_{0}^{4} x(\sqrt{x}-1)^2 dx\]
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15C - Example 5: Video Example
15C - Example 5: Practice
Question 1: Evaluate the following definite integral \[\int_{0}^{\pi}sin(x)dx\] Question 2: Evaluate the following definite integral \[\int_{1}^{2}e^xdx\] 15C - Example 5: Solutions
Question 1: \[\int_{0}^{\pi}sin(x)dx=2\] Question 2: \[\int_{1}^{2}e^xdx=e^2-e\] |