Exam 12: Integration and Its Applications

Use the Midpoint Rule n = 4 to approximate the area of the following region. $f ( y ) = \frac { 1 } { 4 } y , [ 2,4 ]$

Evaluate the integral $\int \left( x ^ { 6 } + 7 x ^ { 4 } - 3 \right) d x$ .

Find the indefinite integral of the following function and check the result by differentiation. $\int \frac { 5 x } { \left( x ^ { 2 } + 5 \right) ^ { 4 } } d x$

Use the values $\int _ { 0 } ^ { 2 } f ( x ) d x = 8$ and $\int _ { 0 } ^ { 2 } g ( x ) d x = 3$ to evaluate the definite integral $\int _ { 0 } ^ { 2 } [ f ( x ) - 2 g ( x ) ] d x$ .

Evaluate the integral $\int \left( - 2 x ^ { 2 } - 9 x - 9 \right) d x$ .

A ball is thrown vertically upwards from a height of 10 ft with an initial velocity of 70 ft per second. How high will the ball go?

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

Find the indefinite integral $\int 4 d x$ and check your result by differentiation.

Find the average value of the function over the given interval. $f ( x ) = 7 \sqrt [ 3 ] { x }$ on [0,8]

Find the indefinite integral. $\int \frac { 7 x ^ { 2 } } { 9 - 8 x ^ { 3 } } d x$

Two models, $R _ { 1 } = 7.21 + 0.58 t$ and $R _ { 2 } = 7.21 + 0.43 t$ , are given for revenue (in billions of dollars per year) for a large corporation. Both models are estimates of revenues for 2007 through 2011, with t = 7 corresponding to 2007. Which model is projecting the greater revenue? How much more total revenue does that model project over the five-year period?

Use the rectangles to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $f ( x ) = - 2 x + 3 , [ 0,1 ]$

Evaluate the integral $\int 210 e ^ { 0.1 x } d x$

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { - 3 } ^ { 3 } \sqrt { 9 - x ^ { 2 } } d x$