91Ó°ÊÓ

Calculate the consumers'surplus at the indicated unit price \(\bar{p}\) for each of the demand equations. $$p=100-3 \sqrt{q} ; \bar{p}=76$$

Decide whether or not the given integral converges. If the integral converges, compute its value. $$\int_{-2}^{+\infty} e^{-0.5 x} d x$$

Find the general solution of each differential equation in Exercises \(1-10 .\) Where possible, solve for \(y\) as a function of \(x\). $$\frac{d y}{d x}=\frac{x}{y}$$

Calculate the consumers'surplus at the indicated unit price \(\bar{p}\) for each of the demand equations. $$p=10-2 q^{1 / 3} ; \bar{p}=6$$

Evaluate the integrals using integration by parts where possible. $$\int(1-x) e^{x} d x$$

Decide whether or not the given integral converges. If the integral converges, compute its value. $$\int_{1}^{+\infty} \frac{1}{x^{1.5}} d x$$

Find the general solution of each differential equation in Exercises \(1-10 .\) Where possible, solve for \(y\) as a function of \(x\). $$\frac{d y}{d x}=\frac{y}{x}$$

Find the averages of the functions over the given intervals. Plot each function and its average on the same graph (as in Figure 8). HINT [See Quick Example page 1033.] \(f(x)=x^{3}-x\) over \([0,1]\)

Find the averages of the functions over the given intervals. Plot each function and its average on the same graph (as in Figure 8). HINT [See Quick Example page 1033.] \(f(x)=e^{-x}\) over \([0,2]\)

Decide whether or not the given integral converges. If the integral converges, compute its value. $$\int_{-\infty}^{2} e^{x} d x$$