Definite Integral:

Evaluate each definite integral.

1. (a) \(\displaystyle{ \int_0^1  (1+x^9+x^{99})\: dx}\)    (b) \(\displaystyle{ \int_0^2  (3x^2-1)\: dx}\)   (c) \(\displaystyle{ \int_1^8 \sqrt[3]{x}\: dx}\)   solution

2. (a) \(\displaystyle{\int_{\ln 2}^{\ln 7}  e^{x} \:dx}\)   (b) \(\displaystyle{ \int_0^{\ln 5} (e^x-1) \: dx}\)    (c) \(\displaystyle{ \int_{-1}^0 (2x-e^x+1) \: dx} \)    solution

3. (a)\(\displaystyle{\int_1^{e^2}  \frac{5}{x} \:dx}\)    (b) \(\displaystyle{ \int_1^9 \frac{x-1}{\sqrt{x}} \: dx}\)   solution

4. A division of Ditton Industries manufactures a deluxe toaster oven. Management has determined that the daily marginal cost function associated with producing these toaster ovens is given by \[C'(x) = 0.0006 x^2 − 0.14x + 22\] where \(C'(x)\) is measured in dollars per unit and \(x\) denotes the number of units produced. Management has also determined that the daily fixed cost incurred in the production is $900. (a) Find the total cost incurred by Ditton in producing the first 300 units of these toaster ovens per day. (b) What is the total cost incurred by Ditton in producing the 201st through 300th units/day?  solution


5. According to a study conducted by the Centers for Medicare and Medicaid Services in 2010, the national spending for out-of-pocket health-care costs is projected to increase ove the next several years. The amount spent annually from \(2010 (t=0)\) is expected to grow at the rate of

\[ R(t)=1.0952t + 1.357 \qquad (0\le t \le 6) \]

billion dollars/year in year \(t\). The national spending in 2010 was $317 billion. What is the projected national spending in 2016?   solution


6. Annual sales (in millions of units) of a certain brand of tablet computers are expected to grow in accordance with the function

\[ f(t) = 0.18 t^2 + 0.16t + 2.64\qquad (0 \le t \le 6) \]

per year, where \( t\) is measured in years. How many tablet computers will be sold over the next 6 years?   solution


7. A division of a certain toy company manufactures an electronic football game. An efficiency study showed that the rate at which the games are assembled by the average worker t hr after starting work at 8 A.M. is \(\displaystyle{-1.5t^2+8t+30, \quad (0 \le t \le 4)}\) units per hour. (a) Find the total number of games the average worker can be expected to assemble in the 4 hr morning shift. (b) How many units can the average worker be expected to assemble in the first hour of the morning shift? (c) How many units can the average worker be expected to assemble in the second hour of the morning shift?  solution

8. Compute the following definite integral.

  1. \(\displaystyle{ \int_{-1}^1  x\,(x^2+1)^2  \, \,  dx}\)  Solution 

  2. \(\displaystyle{ \int_0^1  x \, \sqrt{5x^2+4}  \, \, dx}\)  Solution 

  3. \(\displaystyle{ \int_1^2  \frac{x}{(x^2+3)^2}  \, \,   dx}\)   Solution 

  4. \( \displaystyle{\int_0^1 \frac{x+2}{\sqrt{x^2+4x}}  \, \,    dx}\)    Solution

  5. \( \displaystyle{\int_0^4 \frac{e^{\sqrt{x}}} {\sqrt{x}}   \, \,   dx} \)   Solution