Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.

$$y=e^{x};y=0;0\leq{x}\leq1$$

A trigonometric ratio is

$$\frac { 7 } { 5 }$$

. What ratio could it be, and what angle might it be referring to?

Find the exact value of the logarithmic expression without using a calculator.

Find the domain of the tangent function.

Evaluate each integral. Graph the region of integration, reverse the order of integration, and then evaluate the integral with the order reversed. $\int_{0}^{4} \int_{x / 4}^{\sqrt{x} / 2} x d y d x$

Evaluate each integral. Graph the region of integration, reverse the order of integration, and then evaluate the integral with the order reversed. $\int_{0}^{2} \int_{0}^{y}(y-x)^{4} d x d y$

Evaluate each integral. Graph the region of integration, reverse the order of integration, and then evaluate the integral with the order reversed. $\int_{0}^{4} \int_{y^{2} / 4}^{2 \sqrt{y}}(1+2 x y) d x d y$

Find each antiderivative. Then use the antiderivative to evaluate the definite integral. (A) $\int \frac{\ln x}{x y} d y$ (B) $\int_{1}^{e^{2}} \frac{\ln x}{x y} d y$

Find each antiderivative. Then use the antiderivative to evaluate the definite integral. (A) $\int \frac{\ln x}{x y} d x$ (B) $\int_{1}^{e} \frac{\ln x}{x y} d x$

The shelf life (in years) of a laser pointer battery is a continuous random variable with probability density function

$$f(x)=\begin{cases} 1/(x+1)^{2} &\text{if } x\geq0 \\ 0 &\text{otherwise } \end{cases}$$

(A) Find the probability that a randomly selected laser pointer battery has a shelf life of $3$ years or less.

(B) Find the probability that a randomly selected laser pointer battery has a shelf life of from $3$ to $9$ years.

(C ) Graph $y=f(x)$ for $[0, 10]$ and show the shaded region for part (A).

Evaluate the given definite integral.

$$\displaystyle\int_{0}^{7}\frac{x}{(3+x)(1+x)}\ dx$$

This problem is mixed, some require integration by parts, and others can be solved with techniques considered earlier. Integrate as indicated, assuming $x>0$ whenever the natural logarithm function is involved.

$$\displaystyle\int \frac{\ln{x}}{x}\ dx$$

Evaluate the integral.

$$\displaystyle\int_{0}^{3}\frac{x^{2}}{\sqrt{x^{2}+16}}\ dx$$

The life expectancy (in years) of a microwave oven is a continuous random variable with probability density function

$$f(x)=\begin{cases} 2/(x+2)^{2} &\text{if } x\geq0 \\ 0 &\text{otherwise } \end{cases}$$

(A) Find the probability that a randomly selected microwave oven lasts at most $6$ years.

(B) Find the probability that a randomly selected microwave oven lasts from $6$ to $12$ years

(C ) Graph $y=f(x)$ for $[0, 12]$ and show the shaded region for part (A).