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Integrating Even and Odd Functions: Definite Integrals

Related Topics

Wize University Calculus 1 Textbook > Integration Techniques

Integrating Even and Odd Functions

3 Activities

Wize University Calculus 1 Textbook > Integrals

Computing Definite Integrals Geometrically

3 Activities

Evaluate ∫−114−(x+1)2+sin⁡x⋅e2x4 dx\int_{-1}^1\sqrt{4-\left(x+1\right)^2}+\sin x\cdot e^{2x^4}\ dx.
More Integrating Even and Odd Functions Questions:
Practice: Integration Technique (~F2017 Final Q27)
Evaluate ∫−π2π2cos⁡2x sin⁡x dx\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^2x\ \sin x\ dx.
Practice: Integration Technique (~F2017 Final Q27)
Practice: Integration Technique
Evaluate ∫−π2π2cos⁡2x sin⁡x dx\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^2x\ \sin x\ dx.
Property of Definite Integral
Suppose f(−x)=−f(x)f(-x)=-f(x) and g(−x)=g(x)g\left(-x\right)=g\left(x\right) for all xx values.
If ∫01f(x)dx=5  and  ∫−10g(x)dx=7\displaystyle\int_0^1f\left(x\right)dx=5\ \ \text{and}\ \ \int_{-1}^0g\left(x\right)dx=7, evaluate ∫−11[2f(x)+3g(x)]dx\displaystyle \int_{-1}^1\left[2f\left(x\right)+3g\left(x\right)\right]dx
Integrating Even and Odd Functions
Evaluate ∫−π2π2cos⁡2xsin⁡x dx\displaystyle\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^2x\sin x \ dx
Evaluate ∫−7π37π3sin⁡x dx\displaystyle\int_{-\frac{7\pi}{3}}^{\frac{7\pi}{3}}\sin x\ dx
Evaluate∫−7070x3+x5 dx\text{Evaluate} \displaystyle \int_{-70}^{70} x^3+x^5\space dx
Integrating Even or Odd Functions
Evaluate
∫−ππ(2sin⁡3x−4cos⁡x) dx.\int_{-\pi}^{\pi} (2\sin^3x - 4\cos x)\ dx.
Integrating Even and Odd Functions
Evaluate ∫−π2π2cos⁡2xsin⁡x dx\displaystyle\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^2x\sin x \ dx
Suppose that f(x)f(x) is even, ∫06f(x)dx=9\int_{0}^{6}f(x)dx=9 and ∫−33f(x)dx=6\int_{-3}^{3}f(x)dx=6 and g(x)g(x) is odd with ∫−36g(x)dx=3\int_{-3}^{6}g(x)dx=3. What is the value of the integral ∫362f(x)−3g(x)dx\int_{3}^{6}2f(x)-3g(x)dx?
Evaluate ∫−7π37π3sin⁡x dx\displaystyle\int_{-\frac{7\pi}{3}}^{\frac{7\pi}{3}}\sin x\ dx
Integrating Even and Odd Functions
Suppose that f(−x)=f(x)f\left(-x\right)=f\left(x\right) and ∫055⋅f(x)dx=9\int_0^55\cdot f\left(x\right)dx=9. Find ∫−553⋅f(x) dx\int_{-5}^53\cdot f\left(x\right)\ dx.
Property of Definite Integral
Suppose f(−x)=−f(x)f(-x)=-f(x) and g(−x)=g(x)g\left(-x\right)=g\left(x\right) for all xx values.
If ∫01f(x)dx=5  and  ∫−10g(x)dx=7\displaystyle\int_0^1f\left(x\right)dx=5\ \ \text{and}\ \ \int_{-1}^0g\left(x\right)dx=7, evaluate ∫−11[2f(x)+3g(x)]dx\displaystyle \int_{-1}^1\left[2f\left(x\right)+3g\left(x\right)\right]dx
Practice: Integration Technique (~F2017 Final Q27)
Practice: Integration Technique
Evaluate ∫−π2π2cos⁡2x sin⁡x dx\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^2x\ \sin x\ dx.
More Computing Definite Integrals Geometrically Questions:
Practice: Circle
Q:\textbf{Q:} Find ∫244−(x−2)2\displaystyle\int_2^4\sqrt{4-\left(x-2\right)^2}
Definite Integrals
Calculate
∫01∣x−1∣dx\int_0^1|x-1|dx
(Hint: Sketch a Graph)
Practice: Circle
Q:\textbf{Q:} Find ∫244−(x−2)2\displaystyle\int_2^4\sqrt{4-\left(x-2\right)^2}
Integrals: Geometric Definitions
Evaluate the definite integral ∫5−65+6−x2+10x−19  ⁣dx\displaystyle\int_{5-\sqrt{6}}^{5+\sqrt{6}}\sqrt{-x^2+10x-19}\de{x}.
Evaluate ∫−111−x2dx\int_{-1}^1\sqrt{1-x^2}dx.
Hint: x2+y2=r2x^2+y^2=r^2 is a circle of radius r centered at (0,0)(0,0). Try to draw the curve in this integral.