Podcast
Questions and Answers
Consider the region bounded by $y = \sin(x)$ between $x = 0$ and $x = \pi$. If this region is revolved around the x-axis, what integral represents the volume of the resulting solid?
Consider the region bounded by $y = \sin(x)$ between $x = 0$ and $x = \pi$. If this region is revolved around the x-axis, what integral represents the volume of the resulting solid?
$V = \pi \int_0^{\pi} \sin^2(x) dx$
What is the average value of the function $f(x) = \sqrt{x}$ on the interval $[0, 9]$?
What is the average value of the function $f(x) = \sqrt{x}$ on the interval $[0, 9]$?
2
For the integral $\int x^2 e^{-x} dx$, what is the result after the first application of integration by parts?
For the integral $\int x^2 e^{-x} dx$, what is the result after the first application of integration by parts?
$-x^2e^{-x} + \int 2xe^{-x} dx$
When evaluating $\int e^x \sin(x) dx$ using integration by parts twice, what do you need to do after the second integration by parts to solve for the integral?
When evaluating $\int e^x \sin(x) dx$ using integration by parts twice, what do you need to do after the second integration by parts to solve for the integral?
For the integral $\int \sin^2(x) \cos^5(x) dx$, what substitution should be used to evaluate the integral?
For the integral $\int \sin^2(x) \cos^5(x) dx$, what substitution should be used to evaluate the integral?
Before performing partial fraction decomposition on the integrand $\frac{7x^3 - 3x^2 + 12x - 4}{x^4 + 4x^2}$, what initial step should be taken?
Before performing partial fraction decomposition on the integrand $\frac{7x^3 - 3x^2 + 12x - 4}{x^4 + 4x^2}$, what initial step should be taken?
For $\int \arctan(x) dx$, what would be suitable choices for $u$ and $dv$ when using integration by parts?
For $\int \arctan(x) dx$, what would be suitable choices for $u$ and $dv$ when using integration by parts?
When evaluating the integral $\int \frac{1}{x^2 \sqrt{x^2 - 4}} dx$, what trigonometric substitution is most appropriate?
When evaluating the integral $\int \frac{1}{x^2 \sqrt{x^2 - 4}} dx$, what trigonometric substitution is most appropriate?
Flashcards
Disk Method (Volume)
Disk Method (Volume)
The volume of a solid obtained by rotating a region around the x-axis.
Shell Method (Volume)
Shell Method (Volume)
The volume of a solid obtained by rotating a region around the y-axis.
Average Value of a Function
Average Value of a Function
Integral(f(x)) / (b-a); represents the average height of the function f(x) over [a, b].
Integration by Parts
Integration by Parts
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Trigonometric Integrals
Trigonometric Integrals
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Partial Fraction Decomposition
Partial Fraction Decomposition
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Common Trig Integral
Common Trig Integral
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Arctangent Function
Arctangent Function
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Study Notes
- This is a 60-minute Calculus II practice exam.
- No books, notes, calculators, phones, or other devices are allowed during the exam.
- Students violating this policy will receive a zero on the exam.
- Show your work and provide reasoning for all answers.
- Adhere to the Lehigh University Student Code of Conduct.
Revolving a Region Around the x-axis and y-axis
- Let R be the region under y = sin(x), between x = 0 and x = π.
- Compute the volume V of the solid obtained by revolving the region R about the x-axis.
- Compute the volume V of the solid obtained by revolving the region R about the y-axis.
Average Value of a Function
- Compute the average value of the function f(x) = √x on the interval [0, 9].
- Find all values of c in the interval [0, 9] at which f(c) equals the average value.
Integrals to Evaluate
- Evaluate the integral from 0 to 1 of x²e^(-x) dx.
- Evaluate the integral of e^x sin(x) dx.
Trigonometric Integrals
- Compute the integral of sin²(x) cos³(x) dx.
- Compute the integral of tan³(x) sec³(x) dx.
Integrals with Radicals
- Evaluate the integral of 1 / (x²√(x² + 4)) dx.
Rational Function Integral
- Evaluate the integral of (7x³ - 3x² + 12x - 4) / (x⁴ + 4x²) dx.
Inverse Trigonometric Integral
- Evaluate the integral of arctan(x) dx.
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Description
This Calculus II practice exam includes problems on revolving a region around the x-axis and y-axis, average value of a function, integrals to evaluate, and trigonometric integrals. Show your work and provide reasoning for all answers. No external resources are allowed during the exam.
