Download Math 106 BC Exam 01 - Integration and Volume Calculation and more Exams Calculus in PDF only on Docsity!
The graph of f(x). The graph of Af (x).
- Consider the function f shown above. Next to it, sketch the graph of Af (x) =
∫ (^) x
2
f(t) dt. Make sure the slopes on
your graph are correct. (The curved portion of f is one-quarter of a circle of radius 1 centered at (5,0)).
2A. Use the method of substitution to find the following:
0
e−^2 x √ 1 + e−^2 x^
dx. Show all your steps. Express the answer as
a decimal number to four places after the decimal.
2B. Just to be clear: What are the new limits (written to four decimal places) on the integral after the substitution is made?
- The region S below is bounded by the graphs of y = 1 +
x, y =
(x − 4) and x = 4.
Hint: One of the following problems may require two separate integrals! 3A. Suppose the region S is rotated around the line y = 10. Set up the integral (or integrals) giving the exact volume of the resulting solid of revolution.
3B. Suppose the region S is rotated around the y axis. Set up the integral (or integrals) giving the exact volume of the resulting solid of revolution.
5A. Let I be the exact value of
∫ (^) b
a
f(x) dx where f is some continuous function on [a, b]. “Theorem 3” (from section 6.2)
says that the error committed by MID(n) in approximating I, that is, |I − MID(n)|, is smaller than what expression? (Be sure to say what K 2 means in your answer).
5B. Now let f(x) =
x^4 −
x^3 − 2 x^2 ; then f′^ (x) =
x^3 − 2 x^2 − 4 x and f′′(x) = x^2 − 4 x − 4.
The exact value of
1
f(x) dx is − 1349 /20 = − 67 .45; you do not have to check this. What is the smallest value of n for which theorem 3 guarantees |I − MID(n)| < 0 .0005? Show your calculations. (Use the best possible K 2 ).
5C. For your value of n in the previous problem, what is MID(n) and how far is it from the exact value? Write both answers to six places after the decimal point.
MID(n) equals? the error is?
5D. What is TRAP(30) for this integral? Show any intermediate values needed to find TRAP(30).
