91Ó°ÊÓ

Use the Trapezoidal Rule and simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $$ \int_{0}^{1}\left(\frac{x^{2}}{2}+1\right) d x, n=4 $$

Use the Trapezoidal Rule and simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $$ \int_{0}^{2}\left(x^{4}+1\right) d x, n=4 $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1 / 2}^{\infty} \frac{1}{\sqrt{2 x-1}} d x $$

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .)\) $$ \int_{0}^{1} e^{-x^{2}} d x $$

Use the error formulas to find \(n\) such that the error in the approximation of the definite integral is less than 0.0001 using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{1}^{3} \frac{1}{x} d x $$

Revenue A company sells a seasonal product. The revenue \(R\) (in dollars per year) generated by sales of the product can be modeled by \(R=410.5 t^{2} e^{-t / 30}+25,000, \quad 0 \leq t \leq 365\) where \(t\) is the time in days. (a) Find the average daily receipts during the first quarter, which is given by \(0 \leq t \leq 90\). (b) Find the average daily receipts during the fourth quarter, which is given by \(274 \leq t \leq 365\). (c) Find the total daily receipts during the year.

Present Value A company expects its income \(c\) during the next 4 years to be modeled by \(c=150,000+75,000 t\) (a) Find the actual income for the business over the 4 years. (b) Assuming an annual inflation rate of \(4 \%,\) what is the present value of this income?