Question 1

A cookie company determines that the marginal cost of the xth cookie is given by $$C'(x) = -0.0002x + 1.25$$, with $$C(0) = 0. $$What is the total cost of producing 1000 cookies?

Accepted Answer

$$C(1000) = \$$int_0$$^{1000} (-0.0002x + 1.25)\,dx = 1150$$

Question 2

Find the area under the curve $$y = 5x - x^2$ - 6$$ over the interval $$[2, 3].$$

Accepted Answer

$$\$$int_2$$^3 (5x - x^2$ - 6)\,dx = 1.5$$

Question 3

Evaluate $$\int 7x\,dx.$$

Accepted Answer

$$\frac{7}{2}x^2$ + C$$

Question 4

A transcriptionist’s speed over a 5-min interval is given by $$W'(t) = -5t^2$ + 20t + 30$$, $$t in$$ $$[0, 5]. $$How many words are transcribed during the third minute (from $$t = 2 to$$ $$t = 3$$)?

Accepted Answer

$$\$$int_2$$^3 (-5t^2$ + 20t + 30)\,dt = 60$$

Question 5

Evaluate $$\int 200x^4$\,dx.$$

Accepted Answer

$$40x^5 + C$

Question 6

Find the average value of $$y = 3t^5$ + 2t$$ over $$[-2, 3].$$

Accepted Answer

$$\frac{1}{5} \int_{-2}^3 (3t^5$ + 2t)\,dt = 97.2$$

Question 7

Evaluate $$\int e$$^{-0.3x}$$\,dx$$ using substitution.

Accepted Answer

$$-\frac{1}{0.3}e$$^{-0.3x}$$ + C$$

Question 8

A particle is initially at the origin. Its velocity at any time $$t \geq 0 is$$ $$v(t) = 3t^2$ + 2t. $$Find the distance the particle travels in the first 4 seconds.

Accepted Answer

$$\$$int_0$$^4 (3t^2$ + 2t)\,dt = 80$$

Question 9

Evaluate $$\int x^8$ e$$^{x^9}$$\,dx.$$

Accepted Answer

$$\frac{1}{9} e$$^{x^9}$$ + C$$

Question 10

Evaluate $$\int_{1}^{8} 2x\,dx.$$

Accepted Answer

$$63$$

Question 11

Evaluate $$\int x$$^{32}$$ + e^x + 4x\,dx.$$

Accepted Answer

$$\frac{1}{33}x$$^{33}$$ + e^x + 2x^2$ + C$$

Question 12

Evaluate $$\int_{0}^{6} e$$^{-5x}$$\,dx.$$

Accepted Answer

$$\frac{1}{5}(1 - e$$^{-30}$$)$$

Question 13

Decide whether $$\int_a^b f(x)\,dx is $$positive, negative, or zero if $$f(x) is $$below the x-axis for all $$x in$$ $$[a, b].$$

Accepted Answer

The integral is negative.

Question 14

Evaluate $$\int x^4$ (x^5$ + 5)\,dx$$ using substitution.

Accepted Answer

$$\frac{1}{5}(x^5$ + 5)^2$ + C$$

Question 15

Evaluate $$\int_{1}^{a} 2x\,dx.$$

Accepted Answer

$$a^2 - 1$

Question 16

Integrate $$\int [8(\ln x)^3$ - 2(\ln x)^2$ - 6]x\,dx.$$

Accepted Answer

$$2(\ln x)^4$ - (\ln x)^3$ - 3x + C$$

Question 17

Integrate $$\int (x + 9x + 2)\,dx.$$

Accepted Answer

$$10x^2 + 2x + C$

Question 18

Evaluate $$\int -125x + 4x^3$\,dx.$$

Accepted Answer

Question 19

Approximate $$\$$int_0$$^4 x^2$ + 10\,dx by $$computing the area of each rectangle and adding, using 4 rectangles of equal width.

Accepted Answer

The approximation is $$68$$

Business Calculus Integral Applications and Techniques Review

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