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Math 126 Test II: Integration, Sketching, and Volumes, Exams of Calculus

Acharya Nagarjuna UniversityCalculus

A math test for a university-level calculus course, covering topics such as integral convergence, comparison theorem, sketching regions, finding areas, and rotating regions to find volumes. Students are required to solve problems involving definite integrals, use the comparison theorem, sketch regions, and find the length of a curve.

Typology: Exams

2012/2013

Uploaded on 03/20/2013

shreya
shreya 🇮🇳

4.2

(26)

170 documents

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Math 126 TEST II
Do not use any books or notes. You can use a calculator, but not graphing calculator. If you use a
calculator, leave your results in exact form instead of decimal form. Show all work for full credit.
1. Determine whether each integral is convergent or divergent. Evaluate those that are conver-
gent. (24 points)
(a) Z0
−∞
1
2x−5dx (b) Z∞
1
ln x
x2dx (c) Z1
−1
1
x3dx
pf3
pf4

Partial preview of the text

Download Math 126 Test II: Integration, Sketching, and Volumes and more Exams Calculus in PDF only on Docsity!

Math 126 TEST II

Do not use any books or notes. You can use a calculator, but not graphing calculator. If you use a calculator, leave your results in exact form instead of decimal form. Show all work for full credit.

  1. Determine whether each integral is convergent or divergent. Evaluate those that are conver- gent. (24 points)

(a)

∫ (^0) −∞

2 x − 5 dx^ (b)

∫ (^) ∞ 1

ln x x^2 dx^ (c)

∫ (^1) − 1

x^3 dx

  1. Use the Comparison Theorem to determine whether the integral is convergent or divergent. (16 points) (a)

∫ (^) ∞ 1

cos^2 x 1 + x^2 dx^ (b)

∫ (^) ∞ 1

√^1

x^3 + 1 dx

  1. Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. (20 points)

(a) y = x + 1, y = 9 − x^2 , x = − 1 , x = 2. (b) x + y^2 = 2, x + y = 0.

  1. Eliminate the parameter to find a Cartesian equation of the curve. Then sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. (10 points)

(a) x = 4 cos θ, y = 5 sin θ, −π 2 ≤ θ ≤ π 2 , (b) x = et, y = e−t

  1. Find the exact length of x = y^3 /^2 , 0 ≤ y ≤ 1. (10 points)