Exam 5: Integrals

A car's velocity v (t) at time t is given by the graph below: Use the graph to estimate how far the car travels (a) during the first 10 seconds.(b) on the interval from 10 to 15 seconds.(c) on the interval from 0 to 15 seconds.

Suppose using n = 10 to approximate the integral of a certain function by Simpson's Rule results in an upper bound for the error equal to $\frac { 1 } { 10 }$ . What will the upper bound become if we change to n = 20?

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.(a) $\int _ { 0 } ^ { 1 } \frac { 1 } { ( x - 1 ) ^ { 2 } } d x$ (b) $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 - x } } d x$ (c) $\int _ { 0 } ^ { 4 } \frac { 1 } { ( x - 2 ) ^ { 2 / 3 } } d x$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.(a) $\int _ { - \infty } ^ { 2 } \frac { 1 } { x ^ { 2 } + 4 } d x$ (b) $\int _ { 1 } ^ { \infty } \frac { 1 } { x ^ { 1.0001 } } d x$ (c) $\int _ { 1 } ^ { \infty } \frac { 1 } { x ^ { 09999 } } d x$

Let f (x) = $\frac { 1 } { x }$ on the interval [1,2]. Let the interval be divided into two equal subintervals. Find the value of the Riemann sum $\sum _ { i = 1 } ^ { 2 } f \left( x _ { i } ^ { * } \right) \Delta x _ { i }$ if each $x _ { i } ^ { * }$ is the midpoint of its subinterval.

Estimate $\int _ { 1 } ^ { 3 } \frac { 1 } { x ^ { 2 } } d x$ using the Trapezoidal Rule with n = 4. Then use the error bound $\left| E _ { T } \right| \leq \frac { k ( b - a ) ^ { 3 } } { 12 n ^ { 2 } }$ to estimate the accuracy.

Find the value of the integral $\int _ { 0 } ^ { x / 6 } \frac { \sin x } { \cos ^ { 3 } x } d x$

Evaluate the following integrals: (a) $\int ( x + \sin x ) ^ { 2 } d x$ (b) $\int x e ^ { x ^ { 2 } + 1 } d x$ (c) $\int _ { 0 } ^ { x / 2 } \cos x \sin ^ { 3 } x d x$ (d) $\int \frac { 7 x - 23 } { x ^ { 2 } - 7 x + 12 } d x$

The graph of f below consists of line segments and semicircles. Use it to evaluate the following integrals. (a) $\int _ { 0 } ^ { 14 } f ( x ) d x$ (b) $\int _ { 2 } ^ { 10 } f ( x ) d x$

Find the value of the integral $\int _ { 0 } ^ { \ln 9 } e ^ { x / 2 } d x$

Evaluate $\int \cos ( \ln x ) d x$

Evaluate the following integrals: (a) $\int _ { 0 } ^ { x / 2 } \sin ^ { 4 } x \cos x d x$ (b) $\int _ { 0 } ^ { \pi / 2 } \sin ^ { 4 } x \cos ^ { 3 } x d x$ (c) $\int _ { 0 } ^ { x / 4 } \sin 2 x \cos ^ { 4 } 2 x d x$ (d) $\int _ { 0 } ^ { x } \cos ^ { 4 } x d x$

If $\int _ { 0 } ^ { 1 } f ( x ) d x = 7 \text { and } \int _ { 0 } ^ { 3 } f ( x ) d x = 4 \text {, find the value of } \int _ { 1 } ^ { 3 } f ( x ) d x \text {. }$

Find the value of the integral $\int _ { 0 } ^ { 1 } t ( 2 t - 1 ) ^ { 4 } d t$

Which one of the following is not an improper integral?

Find the value of the integral $\int _ { 0 } ^ { x / 2 } \cos x \sin ( \sin x ) d x$

The velocity of a particle moving along a line is 1 - 2t meters per second. Find the displacement in meters of the particle during the time interval $0 \leq t \leq 1$ .

Find the value of the integral $\int _ { 0 } ^ { 1 } x \tan ^ { - 1 } x d x$

Estimate $\int _ { 1 } ^ { 3 } \frac { 1 } { x ^ { 2 } } d x$ using Simpson's Rule with n = 4. Then use the error bound $\left| E _ { S } \right| \leq \frac { k ( b - a ) ^ { 5 } } { 180 n ^ { 4 } }$ to estimate the accuracy.

Find the value of the integral $\int _ { 0 } ^ { 1 } ( x + 1 ) ^ { 2 } d x$