Math 113, Calculus II
Test 2: Review
1
1.1
Integration
Improper Integrals
Improper integrals arise when an infinity or an undefined point appears either in the integrand, the interval, or the integral. Here's what we do:
a f (x)dx
= limb
b a f (x)
dx dx
Math 113, Calculus II
Winter 2007
Final Exam - Solutions
1. (25 points) Use the limit definition of the definite integral and the sum formulas1 to compute
3 1
3x2 - 4x + 1 dx.
Then check your answer using the Evaluation Theorem. Solution: In this problem,
Math 113, Calculus II
Test 2 - Solutions
1. (25 points) Consider the region bounded by the curves y = x2 and y = 2x - x2 . (a) Sketch the curves and shade in the region bounded by them. 2
y = x2
1 y = 2x - x2
-1 -1
1
2
(b) Find the area of the region abov
Math 113, Calculus II
Final Exam Solutions
1
1. (25 points) Use the limit definition of the definite integral and the sum formulas
2 -2
to compute
4 - x2 dx.
Then check your answer using the Evaluation Theorem. (2) - (-2) 4 Solution: In this integral, we
Math 113, Calculus II 1. (25 points) Consider the region bounded by the curves x 2 = y 3 and y = 1. (a) Sketch both curves and shade in the region. 2
Test 2 Solutions
x2 = y 3
1
y=1
-2
-1 -1
1
2
(b) Find the area of the region above. Solution: Observing t
Test 1 - Solutions
Math 113, Calculus II
February 2, 2007
1. (30 points) Use the limit definition of the definite integral and the sum formulas1 to compute
1 -1
3x2 - 4x + 7 dx.
Then check your answer using the Evaluation Theorem. Solution: In this proble
Calculus II Final Exam Problems - Solutions
1
1. Use the definition of the definite integral to evaluate1
1 -1
(3x - x2 ) dx.
Solution: Recall the limit definition of the definite integral:
b n
f (x) dx = lim
a
n
f (x )x i
i=1
where x = b-a , xi = a + ix
Math 113, Calculus II
Solutions for Test 2: Review
1
Practice Problems - Solutions
1. Determine whether the following series are convergent or divergent. Specify the test you use in each part! (a) n2 + 1 2n2 - 1 Solution: Use the Ratio Test as follows. Co
Math 113, Calculus II
Test 1: Review Solutions
1. Use the limit definition of the definite integral to evaluate
5
(4x - x2 )dx.
2
Solution: By definition, 2 (4x - x2 )dx = limn n f (x )x, where x = b-a = 5-2 = i i=1 n n 3 2 n , f (x) = 4x - x and we may t
Math 113, Calculus II
Test 1: Review
1
Review of the material
Chapter 5 is an introduction to integral calculus. In this chapter we study techniques for evaluating both definite and indefinite integrals of continuous functions.
1.1
SIGMA NOTATION
A conven
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Math 113, Calculus II
Test 1 Solutions
1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute
0
x2 +
-3
4 3
dx.
Then check your answer using the Evaluation Theorem. 3 3 0 - (-3) = , so xj = a + j x = -3 + j = Solu