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Problem 1

The velocity graph of a car accelerating from rest to a speed of 120 km/h over a period of 30 seconds is shown. Estimate the distance traveled during this period

Problem 2

Using Definition 2, find an expression for the area under the function f(x) = x 2 e x on the interval 0 ≤ x ≤ 4. Do not evaluate the limit.

2 Definition The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles:

Problem 3

Determine a region whose area is given by

Problem 4

If A is the area under the curve y = e x from 1 to 3, use the inequality to find a value of n such that Rn − A < 0.0001.

Problem 5

Use the midpoint rule with n = 4 to evaluate

Problem 6

Express this sum as a definite integral on [2, 5]:

Problem 7

Show that the definite integral is equal to limn→∞ Rn and then evaluate the limit.

where

Problem 8

The graph of g consists of two straight lines and a semicircle. Use it to evaluate each integral.

(a) g(x) dx

(b) g(x) dx

(c) g(x) dx