Q: A car's velocity at time $$t is $$given by $$v(t) = 4t. $$What is the total distance traveled by the car from $$t=0 to$$ $$t=5$$ seconds?

$$\int_{0}^{5} 4t \, dt = 2t^2$ \Big|_0^5$ = 2 \times 25 - 0 = 50$$

Question 1

Suppose a population grows according to the exponential model $$P(t) = P_0$ e$$^{rt}$$, where P_0$ is the initial population and r$ is the growth rate. If P_0$ = 1000$$ and $$r = 0.03$$, what will the population be after 5 years?

Accepted Answer

$$P(5) = 1000 e$$^{0.15}$$ \approx 1162$$

Question 2

Given the function $$f(x) = x^2$, what is the definite integral of f(x$$)$$ from x=1$ to x=3$?

Accepted Answer

$$\int_{1}^{3} x^2$ \, dx = \left[ \frac{x^3$}{3} \right]_1^3$ = \frac{27}{3} - \frac{1}{3} = \frac{26}{3}$$

Question 3

A car's velocity at time $$t is $$given by $$v(t) = 4t. $$What is the total distance traveled by the car from $$t=0 to$$ $$t=5$$ seconds?

Accepted Answer

$$\int_{0}^{5} 4t \, dt = 2t^2$ \Big|_0^5$ = 2 	imes 25 - 0 = 50$$

Question 4

Which of the following best describes the Fundamental Theorem of Calculus?

Accepted Answer

It relates differentiation and integration, stating that the derivative of the integral of a function is the original function.

Question 5

Evaluate the definite integral $$\int_{0}^{\pi} \sin x \, dx.$$

Accepted Answer

$$\int_{0}^{\pi} \sin x \, dx = [-\cos x]_0$$^{\pi}$$ = -\cos(\pi) + \cos(0) = 2$$

Question 6

If $$f(x) = 3x^2$ + 2x$$, what is the value of $$\int_{2}^{4} f(x) \, dx$$?

Accepted Answer

$$\int_{2}^{4} (3x^2$ + 2x) \, dx = [x^3$ + x^2$]_2^4$ = (64 + 16) - (8 + 4) = 80 - 12 = 68$$

Question 7

A medication is administered at a rate of $$R(t) = 5e$$^{-0.2t}$$ mg/hour. How much medication is administered in the first 10 hours?

Accepted Answer

$$\int_{0}^{10} 5e$$^{-0.2t}$$ \, dt = \left[ -25e$$^{-0.2t}$$ \right]_0$$^{10}$$ = -25e$$^{-2}$$ + 25 = 25(1 - e$$^{-2}$$)$$

Question 8

Which of the following is the correct definition of a definite integral?

Accepted Answer

The definite integral of $$f(x)$$ from $$a to$$ $$b is$$ $$\int_{a}^{b} f(x) \, dx = \lim_{n 	o \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$$

Question 9

Which of the following is a property of definite integrals?

Accepted Answer

$$\int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx$$

Question 10

A tank is being filled with water at a rate of $$r(t) = 10 - t$$ liters per minute. How much water is added to the tank in the first 8 minutes?

Accepted Answer

$$\int_{0}^{8} (10 - t) \, dt = [10t - \frac{t^2$}{2}]_0^8$ = 80 - 32 = 48$$

Question 11

Which of the following statements is true about the area under the curve $$y = f(x)$$ from $$x=a to$$ $$x=b$$?

Accepted Answer

It is given by the definite integral $$\int_{a}^{b} f(x) \, dx$$

Question 12

If $$f(x) is $$continuous on $$[a, b]$$, which theorem guarantees that $$\int_{a}^{b} f(x) \, dx$$ exists?

Accepted Answer

The existence of the definite integral is guaranteed by the Fundamental Theorem of Calculus.

Question 13

A chemical reaction rate is modeled by $$r(t) = 2t^2$. How much reactant is consumed between t=1$ and t=4$?

Accepted Answer

$$\int_{1}^{4} 2t^2$ \, dt = [\frac{2}{3}t^3$]_1^4$ = \frac{2}{3}(64 - 1) = \frac{2}{3} 	imes 63 = 42$$

Question 14

Which of the following is the correct expression for the average value of a continuous function $$f(x) on$$ $$[a, b]$$?

Accepted Answer

$$\frac{1}{b-a} \int_{a}^{b} f(x) \, dx$$

Question 15

A patient receives a drug at a rate of $$r(t) = 6 - 0.5t mg/$$hour. How much drug is administered in the first 6 hours?

Accepted Answer

$$\int_{0}^{6} (6 - 0.5t) \, dt = [6t - 0.25t^2$]_0^6$ = 36 - 9 = 27$$

Question 16

Which of the following is the correct statement of the linearity property of definite integrals?

Accepted Answer

$$\int_{a}^{b} [af(x) + bg(x)] \, dx = a \int_{a}^{b} f(x) \, dx + b \int_{a}^{b} g(x) \, dx$$

Question 17

If $$f(x) is an $$odd function, which of the following is true about $$\int_{-a}^{a} f(x) \, dx$$?

Accepted Answer

$$\int_{-a}^{a} f(x) \, dx = 0$$

Integration Techniques and Properties: Calculus Study Notes

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