Exam 5: Antiderivatives and Indefinite Integration

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { - a } ^ { a } \sqrt { a ^ { 2 } - z ^ { 2 } } d z$

The graph of the function $f ( x ) = 16 - x ^ { 2 }$ is given below. Which of the following definite integrals yields the area of the shaded region?

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { 1 } ^ { 4 } 2 s d s$

$\text { Find the average value of the function } f ( x ) = 48 - 12 x ^ { 2 } \text { over the interval } - 5 \leq s \leq 5 \text {. }$

Find the indefinite integral $\int 11 \tan ^ { 2 } x + 16 d x$

Solve the differential equation. $\frac { d f } { d z } = 4 z + \frac { 3 z } { \sqrt { 4 - z ^ { 2 } } }$

Find the indefinite integral of the following function. $\int \frac { \cos x } { \sin ^ { 8 } x } d x$

$\text { Use a computer algebra system and Simpson's Rule with } n = 10 \text { to approximate } t \text { in }$ the integral equation $\int _ { 0 } ^ { t } \sin \sqrt { x } d x = 3$ . Round your answer to three decimal places.

Find the sum given below. $\sum _ { i = 1 } ^ { 5 } ( 2 i + 4 )$

Use the error formula to estimate the error in approximating the integral $\int _ { 3 } ^ { 6 } ( 9 x + 1 ) d x$ with $n = 4$ using Trapezoidal Rule.

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { 0 } ^ { 9 } \sqrt { 81 - t ^ { 2 } } d t$

Find the indefinite integral of the following function and check the result by differentiation. $\int \frac { 6 s ^ { 5 } } { s ^ { 6 } + 4 } d s$

Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 8 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral. $\int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 1 + s ^ { 3 } } } d s$

The maker of an automobile advertises that it takes $12 \text { seconds to accelerate from } 30$ kilometers per hour to $85$ kilometers per hour. Assuming constant acceleration, compute the acceleration in meters per second per second. Round your answer to three decimal places.

$\text { Evaluate the integral } \int _ { 0 } ^ { 1 / 6 } \frac { 16 } { \sqrt { 1 - 9 x ^ { 2 } } } d x$

Find the indefinite integral. $\int \frac { 2 x - 4 } { x ^ { 2 } + 4 x + 13 } d x$

Write the following limit as a definite integral on the interval $[ 2,8 ] \text { where } c _ { i } \text { is any }$ point in the $i ^ { \text {th } } \text { subinterval. }$ $\lim _ { | \Delta x | \rightarrow 0 } \sum _ { i = 1 } ^ { n } \sqrt { 3 c _ { i } ^ { 2 } + 4 c _ { i } } \Delta x _ { i }$

Evaluate the integral. $\int _ { 7 } ^ { 6 } - 24 x d x$ given, $\int _ { 6 } ^ { 7 } x ^ { 3 } d x = \frac { 1105 } { 4 } ,$ 7dx= 6 xdx= 6 7 dx=1 6

Find the indefinite integral. $\int e ^ { 3 x } \sec \left( e ^ { 3 x } \right) \tan \left( e ^ { 3 x } \right) d x$

Find the indefinite integral and check the result by differentiation. $\int \frac { 3 z ^ { 2 } + 12 z - 9 } { z ^ { 4 } } d z$