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Quizzes > Mathematics & Statistics

Calculus II Quiz

Free Practice Quiz & Exam Preparation

Difficulty: Moderate
Questions: 15
Study OutcomesAdditional Reading
Calculus II course depicted through 3D voxel art in high quality

This Calculus II quiz helps you practice the core ideas you meet in class: advanced integration, conic sections, polar coordinates, and infinite series. Work through 15 quick questions to spot gaps before an exam and build speed with common steps. Your results show what to review next.

Which integration technique is most effective for evaluating ∫ x e^x dx?
Substitution
Trigonometric Substitution
Integration by Parts
Partial Fractions
Integration by parts is ideal for integrals that are the product of an algebraic and an exponential function. This method reduces the original integral into an easier one to evaluate.
Which equation represents a circle centered at the origin with a radius of 5?
x² + 5y² = 25
x² + y² = 25
x²/25 + y²/25 = 1
(x - 5)² + (y - 5)² = 25
A circle centered at the origin with radius r is defined by the equation x² + y² = r². Here, substituting r = 5 gives the correct equation x² + y² = 25.
Express the Cartesian point (0, -5) in polar coordinates, with r > 0 and 0 ≤ θ < 2π.
(5, 3π/2)
(5, π/2)
(5, π)
(-5, π/2)
The distance from the origin for the point (0, -5) is 5 and the angle corresponding to the negative y-axis is 3π/2 when measured in standard polar coordinates. Therefore, the correct polar representation is (5, 3π/2).
Which of the following series is an example of a convergent p-series?
' (1/(n log n))
' (1/√n)
' (1/n)
' (1/n²)
A p-series of the form ' 1/n^p converges when p > 1. Since in ' (1/n²) the exponent p is 2, the series converges, making it the correct choice.
Which substitution simplifies the integral ∫ (2x)/(x² + 1) dx?
Let u = x²
Let u = x² + 1
Let u = x
Let u = 2x
By letting u = x² + 1, the differential becomes du = 2x dx, which exactly matches the numerator of the integrand. This substitution directly simplifies the integral into an elementary form.
Evaluate the integral ∫ (dx/√(9 - x²)).
cos❻¹(x/3) + C
tan❻¹(x/3) + C
-sin❻¹(x/3) + C
sin❻¹(x/3) + C
The integral ∫ (dx/√(a² - x²)) is a standard form with the antiderivative sin❻¹(x/a) + C. Here, a² = 9 implies that a = 3, making sin❻¹(x/3) + C the correct answer.
Evaluate the integral ∫ ((2x + 3)/(x² + x - 2)) dx using partial fractions.
ln| (x + 2)/(x - 1) | + C
(5/3) ln|x + 2| + (1/3) ln|x - 1| + C
(1/3) ln|x + 2| + (5/3) ln|x - 1| + C
2 ln| (x + 2)(x - 1) | + C
Factorizing the denominator as (x + 2)(x - 1) and expressing the integrand in partial fractions gives coefficients A = 1/3 and B = 5/3. Integrating these terms leads to the logarithmic expression provided in the correct answer.
Which substitution is most effective for evaluating the integral ∫ √(x² + 4) dx?
x = 2 cosθ
x = 2 tanθ
x = 2 secθ
x = 2 sinθ
For integrals of the form ∫ √(x² + a²) dx, the substitution x = a tanθ transforms the square root into a secant function, simplifying the integral. Here, with a = 2, the substitution x = 2 tanθ is most effective.
Determine the radius of convergence for the power series ' ((x - 2)❿/(n·3❿)).
1
6
3
2
Applying the ratio test to the series results in a limit of |x - 2|/3, which must be less than 1 for convergence. Therefore, the series converges when |x - 2| < 3, giving a radius of convergence of 3.
Evaluate the integral ∫ tan²(x) dx.
sec²(x) - x + C
-tan(x) + x + C
tan(x) - x + C
tan(x) + x + C
By using the identity tan²(x) = sec²(x) - 1, the integral becomes ∫ (sec²(x) - 1) dx, which simplifies to tan(x) - x + C. This step-by-step approach confirms that the correct antiderivative is tan(x) - x + C.
Identify the conic section and its center for the equation 4x² + 9y² - 16x + 36y - 44 = 0.
Ellipse with center (2, -2)
Circle with center (2, -2)
Hyperbola with center (2, -2)
Parabola with vertex (2, -2)
Completing the square for both x and y terms transforms the given equation into the standard form of an ellipse, revealing that its center is at (2, -2). The differing coefficients for x² and y² confirm its elliptical nature.
Find the area enclosed by one loop of the polar curve r = 2 sin(θ).
π/2
2π
4π
π
The area enclosed by a polar curve is given by (1/2)∫ r² dθ. For r = 2 sin(θ), integrating from 0 to π yields an area of π, which confirms the correct answer.
Determine the convergence of the series ' (from n = 2 to ∞) 1/(n (ln(n))²).
It converges conditionally
It converges
It converges absolutely
It diverges
Using the integral test, the series ' 1/(n (ln(n))²) is found to converge because the corresponding integral converges. This confirms that the series as given converges.
Evaluate the integral ∫ e^x sin(x) dx.
e^x(sin(x) - cos(x)) + C
(e^x/2)(sin(x) - cos(x)) + C
(e^x/2)(sin(x) + cos(x)) + C
-(e^x/2)(sin(x) - cos(x)) + C
Applying integration by parts twice to the integral ∫ e^x sin(x) dx leads to an equation that can be solved for the integral. The resulting antiderivative is (e^x/2)(sin(x) - cos(x)) + C.
Determine whether the series ' (-1)^n / √n converges absolutely, converges conditionally, or diverges.
It converges absolutely
It converges conditionally
It diverges
It converges uniformly
The alternating series ' (-1)^n/√n satisfies the conditions of the alternating series test, which guarantees convergence. However, the absolute series ' 1/√n is a divergent p-series (with p = 1/2 < 1), resulting in conditional convergence overall.
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Study Outcomes

  1. Apply advanced integration techniques to solve complex integrals.
  2. Analyze the properties and equations of conic sections in analytic geometry.
  3. Convert and graph equations using polar coordinates.
  4. Evaluate infinite series and assess their convergence or divergence.

Calculus II Additional Reading

Here are some top-notch resources to supercharge your Calculus II journey:

  1. Dive into MIT's comprehensive lectures covering advanced integration techniques, arc length, surface area, and polar coordinates. Perfect for deepening your understanding with high-quality materials.
  2. Explore Ohio State University's interactive textbook, offering a structured approach to integration techniques, applications, and infinite series, complete with examples and exercises.
  3. Access a treasure trove of lecture notes, homework assignments, and video tutorials from Berkeley City College, covering areas between curves, volumes, work problems, and more.
  4. Practice with a variety of worksheets from the University of Arizona, focusing on integration techniques, applications, sequences, and series to reinforce your learning.
  5. Peruse detailed notes from East Tennessee State University, covering volumes, arc length, surface area, and differential equations, complete with examples and exercises.
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