Exam 6: The Integral

The average value of $f ( x ) = | 2 x - 3 |$ on [0, 3] is

The solution to the differential equation $\frac { d y } { d x } = \frac { e ^ { x } } { 2 y }$ satisfying the boundary condition $y = 1$ when $x = 0$ is

The derivative $\frac { d } { d x } \left[ \int _ { 0 } ^ { \sin x } \sqrt { 1 - t ^ { 2 } } d t \right]$ is

The derivative $\frac { d } { d x } \left[ \int _ { 3 } ^ { x ^ { 2 } + 1 } 2 ^ { t } d t \right]$ is

The derivative $\frac { d } { d x } \left[ \int _ { x ^ { 2 } + 1 } ^ { 4 } \ln \left( \frac { 1 } { t } \right) d t \right]$ is

The solution to the differential equation $\frac { d y } { d x } = 2 x y$ satisfying the boundary condition $y = e ^ { 2 }$ when $x = 0$ is

The solution to the differential equation $\frac { d y } { d x } = \frac { y } { x }$ satisfying the boundary condition $y = 1$ when $x = 1$ is

Let $\int _ { 1 } ^ { 3 } f ( x ) d x = 14$ and $\int _ { 8 } ^ { 3 } f ( x ) d x = - 5$ Then $\int _ { 1 } ^ { 8 } f ( x ) d x$ is

The antiderivative $\int \frac { 1 } { x \sqrt [ 3 ] { x } } d x$ is

If f is an even function, $\int _ { - 4 } ^ { 0 } f ( x ) d x = 4,$ and $\int _ { 0 } ^ { 2 } f ( x ) d x = 5,$ then $\int _ { 2 } ^ { 4 } f ( x ) d x$ is

Let A denote the area enclosed by the graph $f ( x ) = \frac { 1 } { x }$ , the x-axis, and the lines x = 1 and x = e. By part 2 of the Fundamental Theorem of Calculus, A is

If $F ( x ) = \int _ { 1 } ^ { x } ( \sqrt { t } + 1 ) d t$ , what is F(4)?

The indefinite integral $\int x ^ { 2 } e ^ { x ^ { 3 } - 2 } d x$ is

The average value of $f ( x ) = 1 + \sin x$ on $[ 0 , \pi ]$ is

Let A denote the area enclosed by the graph $f ( x ) = | x | \text {, }$ the x-axis, and the lines $x = - 1$ and $x = 1$ . Graphing the region and using plane geometry, we can find that A is

The rate of water consumption (in hundreds of gallons per year) in an office building since its opening in 1995 is modeled by the function $w = \frac { 1 } { 4 } t$ , where tis the number of years after 1995. Which integral represents the total number of gallons consumed between 1996 and 2001?

The antiderivative $\int \left( e ^ { x } + x ^ { e } \right) d x$ is

Suppose S₆ is the upper sum of the area enclosed by the graph $f ( x ) = 10 - x ^ { 2 },$ the x-axis, and the lines x = 0 and x = 3 by partitioning [0, 3] into six subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2], [2, 2.5], and [2.5, 3]. Then S₆ is

The derivative $\frac { d } { d x } \left[ \int _ { x } ^ { 3 } 2 ^ { t } d t \right]$ is

The bounds m and M used in the Bounds on an Integral Theorem for $\int _ { - 1 } ^ { 2 } \left[ 2 - ( x - 1 ) ^ { 2 } \right] d x$ are