Exam 9: Techniques of Integration

Find the indicated probability. - $f ( x ) = \cos x \text { over } \left[ 0 , \frac { \pi } { 2 } \right] , \mathrm { P } \left( \frac { \pi } { 7 } \leq x \leq \frac { \pi } { 3 } \right)$

Evaluate the integral. -Use the formula $\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int f ( y ) d y , y = f ^ { - 1 } ( x )$ to evaluate the integral. $\int \cos ^ { - 1 } x d x$

Use Simpson's Rule with n = 4 steps to estimate the integral. - $\int _ { 1 } ^ { 3 } ( 2 x + 2 ) d x$

Solve the problem. -The amount of time that goes by between a new driver getting a license and the moment the driver is involved in an accident is exponentially distributed. An insurance company observes a sample of new drivers and finds That 60% are involved in an accident during the first 4 years after they get their driver's license. In a group of 300 New drivers, how many should the insurance company expect to be involved in an accident during the first 6.0 Years after receiving their license?

Solve the problem. -Find the area of the region enclosed by the curve $y = x \sin x$ and the $x$ -axis for $5 \pi \leq x \leq 6 \pi$ .

Evaluate the integral by making a substitution and then using a table of integrals. - $\int \tan t \cdot \sqrt { 1 - \cos ^ { 2 } t } d t$

Provide an appropriate response. -The "trapezoidal" sum can be calculated in terms of the left and right-hand sums as $\underline { ? }$ .

Evaluate the integral by using a substitution prior to integration by parts. - $\int ( \ln 2 x ) ^ { 2 } d x$

Evaluate the integral. The integral may not require integration by parts. - $\int x ^ { 4 } \sec x ^ { 5 } d x$

Integrate the function. - $\int _ { - 1 } ^ { 1 } \frac { 4 } { 1 + 16 t ^ { 2 } } d t$

Determine whether the improper integral converges or diverges. - $\int _ { 2 } ^ { \infty } \frac { 4 } { ( x + 1 ) ^ { 2 } } d x$

Integrate the function. - $\int \frac { \left( 4 - t ^ { 2 } \right) ^ { 3 / 2 } } { t ^ { 6 } } d t$

Solve the problem by integration. - $\text { Find the } x \text {-coordinate of the centroid of the area bounded by } y \left( x ^ { 2 } - 9 \right) = 1 , y = 0 , x = 4 \text {, and } x = 7 \text {. }$

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 0 } ^ { 4 } \left( 7 x ^ { 2 } - 3 x \right) d x$ with an error of magnitude less than $10 ^ { - 4 }$ using Simpson's Rule.

Find the surface area or volume. -Use numerical integration with a programmable calculator or a CAS to find, to two decimal places, the area of the surface generated by revolving the curve $y = \cos x , 0 \leq x \leq \frac { \pi } { 2 }$ , about the $x$ -axis.

Evaluate the integral. - $\int \frac { \sqrt { 25 x ^ { 2 } - 9 } } { x } d x$

Evaluate the integral. - $\int \left( x ^ { 2 } - 3 x \right) e ^ { x } d x$

Determine whether the improper integral converges or diverges. - $\int _ { 1 } ^ { \infty } \frac { | \sin x | } { x ^ { 2 } } d x$

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 0 } ^ { 3 } ( 2 x + 6 ) d x$ with an error of magnitude less than $10 ^ { - 4 }$ using the Trapezoidal Rule.

Determine whether the improper integral converges or diverges. - $\int _ { 1 } ^ { \infty } e ^ { - x } \sin x d x$