Evaluate each iterated integral. (Many of these use results from Exercises 1-10 ).$$\int_{0}^{3} \int_{4}^{5} x \sqrt{x^{2}+3 y} d y d x$$

Evaluate each iterated integral. (Many of these use results from Exercises 1-10 ). $$\int_{1}^{2} \int_{4}^{9} \frac{3+5 y}{\sqrt{x}} d x d y$$

Evaluate each iterated integral. (Many of these use results from Exercises 1-10 ). $$\int_{16}^{25} \int_{2}^{7} \frac{3+5 y}{\sqrt{x}} d y d x$$

Evaluate each iterated integral. (Many of these use results from Exercises 1-10 ). $$\int_{1}^{6} \int_{1}^{6} \frac{d x d y}{5 x y}$$

Evaluate each iterated integral. (Many of these use results from Exercises 1-10 ). $$\int_{1}^{5} \int_{2}^{4} \frac{1}{y} d x d y$$

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Evaluate each iterated integral. (Many of these use results from Exercises 1-10 ). $$\int_{0}^{1} \int_{3}^{6} x \sqrt{x^{2}+3 y} d x d y$$

$=\frac{2}{45}\left(39^{5 / 2}-12^{5 / 2}-7533\right)$

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Evaluate each iterated integral. (Many of these use results from Exercises 1-10 ). $$\int_{0}^{3} \int_{1}^{2}\left(x y^{3}-x\right) d y d x$$

In Exercises 47-56, (a) use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant over the intervals you identified in part (a).

$g(x) = x$

Applying the First Derivative Test In Exercises $41-48$ , consider the function on the interval $(0,2 \pi) .(a)$ Find the open intervals on which the function is increasing or decreasing. (b) Apply the First Derivative Test to identify all relative extrema. (c) Use a graphing utility to confirm your results.$f(x)=\frac{\sin x}{1+\cos ^{2} x}$

In Exercises 53-60, (a) use a graphing utility to graph each side of the equation to determine whether the equation is an identity, (b) use the table feature of a graphing utility to determine whether the equation is an identity, and (c) confirm the results of parts (a) and (b) algebraically.

$ \dfrac{1 + \cos x}{\sin x} = \dfrac{\sin x}{1 - \cos x} $

In Exercises $33-36$ , use division to write the rational function in the form $q(x)+r(x) d(x),$ where the degree of $r(x)$ is less than the degree of $d(x) .$ Then find the partial fraction decomposition of $r(x) / d(x) .$ Compare the graphs of the rational function with the graphs of its terms in the partial fraction decomposition.$$\frac{x^{3}+2}{x^{2}-x}$$

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## Recommended Questions

Evaluate each iterated integral. (Many of these use results from Exercises 1-10 ).

$$\int_{0}^{3} \int_{4}^{5} x \sqrt{x^{2}+3 y} d y d x$$

Evaluate each iterated integral. (Many of these use results from Exercises 1-10 ).

$$\int_{0}^{3} \int_{1}^{2}\left(x y^{3}-x\right) d y d x$$

Evaluate each iterated integral. (Many of these use results from Exercises 1-10 ).

$$\int_{16}^{25} \int_{2}^{7} \frac{3+5 y}{\sqrt{x}} d y d x$$

In Exercises 47-56, (a) use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant over the intervals you identified in part (a).

$g(x) = x$

Applying the First Derivative Test In Exercises $41-48$ , consider the function on the interval $(0,2 \pi) .(a)$ Find the open intervals on which the function is increasing or decreasing. (b) Apply the First Derivative Test to identify all relative extrema. (c) Use a graphing utility to confirm your results.

$f(x)=\frac{\sin x}{1+\cos ^{2} x}$

In Exercises 53-60, (a) use a graphing utility to graph each side of the equation to determine whether the equation is an identity, (b) use the table feature of a graphing utility to determine whether the equation is an identity, and (c) confirm the results of parts (a) and (b) algebraically.

$ \dfrac{1 + \cos x}{\sin x} = \dfrac{\sin x}{1 - \cos x} $

In Exercises $33-36$ , use division to write the rational function in the form $q(x)+r(x) d(x),$ where the degree of $r(x)$ is less than the degree of $d(x) .$ Then find the partial fraction decomposition of $r(x) / d(x) .$ Compare the graphs of the rational function with the graphs of its terms in the partial fraction decomposition.

$$\frac{x^{3}+2}{x^{2}-x}$$