Exam 5: Integrals

Evaluate $\int _ { - 1 } ^ { 2 } ( 2 - | x | ) d x$ by interpreting the integral in terms of area.

Evaluate the following integrals: (a) $\int \left( x ^ { 2 } + 1 \right) e ^ { x ^ { 3 } + 3 x } d x$ (b) $\int \sin \sqrt { x } d x$ (c) $\int \frac { x ^ { 2 } + x + 1 } { \sqrt { x } } d x$ (d) $\int \sqrt { 4 + x ^ { 2 } } d x$

The graph of a function f is given below. Estimate $\int _ { 0 } ^ { 8 } f ( x ) d x$ using 4 subintervals with sample points: (a) right endpoints.(b) left endpoints.(c) midpoints.

The following table shows the speedometer readings of a truck, taken at ten minute intervals during one hour of a trip. Time () 0 10 20 30 40 50 60 Speed (/) 40 45 50 60 70 65 60 Use the table and the indicated technique to estimate the distance that the truck traveled in the hour.(a) The Trapezoidal Rule (b) The Midpoint Rule (c) Simpson's Rule

Evaluate $\int \frac { \cos x } { 4 + \sin ^ { 2 } x } d x$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.(a) $\int _ { 1 } ^ { \infty } x ^ { - 5 / 4 } d x$ (b) $\int _ { - \infty } ^ { 1 } \frac { 1 } { \sqrt { x } } d x$ (c) $\int _ { 0 } ^ { \infty } x e ^ { - 2 x } d x$

Use the Midpoint Rule with n = 2 to approximate the integral $\int _ { 0 } ^ { 1 } x ^ { 3 } d x$

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

What is the value of the estimate using four approximating rectangles and taking sample points to be left-hand endpoints?

The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use Simpson's Rule to estimate the area of the pool.

Use the Comparison Theorem to determine whether $\int _ { 0 } ^ { \infty } \frac { 1 } { x + e ^ { x } } d x$ is convergent or divergent. Justify your answer.

Evaluate the following integrals: (a) $\int \left( e ^ { x } + e ^ { - x } \right) ^ { 2 } d x$ (b) $\int \frac { x + 3 } { \left( x ^ { 2 } + 6 x \right) ^ { 2 } } d x$ (c) $\int \frac { x + 3 } { x ^ { 2 } + 6 x } d x$ (d) $\int _ { - x / 2 } ^ { x / 2 } \frac { x \cos x } { 1 + x ^ { 4 } } d x$

Evaluate the following integrals: (a) $\int x \sqrt { x ^ { 2 } - 4 } d x$ (b) $\int \sqrt { x ^ { 2 } - 4 } d x$ (c) $\int \frac { 1 } { x ^ { 2 } + 2 x + 2 } d x$ (d) $\int \frac { x ^ { 3 } } { 1 + x ^ { 2 } } d x$

Use Definition 2 to show that the area under the curve $y = x ^ { 3 } , 0 \leq x \leq 1 \text { is } \frac { 1 } { 4 }$ Recall that $\sum _ { k = 1 } ^ { n } k ^ { 3 } = \frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 }$

Let f (x) = $2 x + 1$ on the interval [ - 1,1]. Let the interval be divided into four equal subintervals. Find the value of the Riemann sum $\sum _ { i = 1 } ^ { 4 } f \left( x _ { i } ^ { * } \right) \Delta x _ { i } \text { if } x _ { i } ^ { * }$ is the right endpoint of its subinterval.

Evaluate the improper integral $\int _ { 0 } ^ { 1 } x ^ { - 2 } d x$

If $\int _ { 0 } ^ { 3 } f ( x ) d x = 12 \text { and } \int _ { 0 } ^ { 6 } f ( x ) d x = 42 \text {, find the value of } \int _ { 3 } ^ { 6 } ( 2 f ( x ) - 3 ) d x \text {. }$

Use the Comparison Properties of Integrals to show that for any $b \geq 0 , \int _ { 0 } ^ { b } 2 ^ { t } d t \geq b$

Let f (x) = $\frac { 1 } { x }$ on the interval [1,2]. Let the interval be divided into four equal subintervals. Find the value of the Riemann sum $\sum _ { i = 1 } ^ { 4 } f \left( x _ { i } ^ { * } \right) \Delta x _ { i }$ if each $x _ { i } ^ { * }$ is the left endpoint of its subinterval.

Evaluate the following integrals: (a) $\int _ { 2 } ^ { 3 } 2 x - 6 x ^ { 2 } d x$ (b) $\int _ { 1 } ^ { 3 } 5 ^ { x } d x$ (c) $\int _ { 1 } ^ { 3 } \frac { x ^ { 2 } - 4 x } { x ^ { 2 } } d x$ (d) $\int _ { 1 } ^ { 7 } \frac { 1 } { \sqrt [ 3 ] { t } } d t$