Antiderivatives

​6.3 Definite Integrals and
Anti-derivatives

​Properties of Definite Integrals

​Properties of Definite Integrals

Multiple Choice

What is the meaning of the following symbol:

$$\int_a^bf\left(x\right)\ dx$$  

• The area from the curve to the x-axis from x = a to x = b

• The rate at which f(x) is changing from x = a to x = b

• The volume when the curve is rotated about the x-axis from x = a to x = b

Multiple Choice

 Evaluate:

• 9

• 11

• -11

Multiple Choice

 Evaluate:

• 9

• 11

• -2

Multiple Choice

 Evaluate:

• 28

• 14

• -2

Multiple Choice

 Evaluate:

• -5

• 23

• -2

Multiple Choice

 Evaluate:

• -5

• 23

• 0

Multiple Choice

 Evaluate:

• 32

• 0

Multiple Choice

 Evaluate:

Multiple Choice

 Evaluate:

​Average Value of a Function

Multiple Choice

f(x)=4-x² over [0,3] find the average value of the function.

• 3

• 12

• 1

Multiple Choice

f(x)=x² -1 , over $$\left[0,\sqrt[]{3}\right]$$ find the average value of the function.

• 3

• 12

• 1

Finite Integral of Trigonometric Functions

Multiple Choice

$$19.\ \int_{\pi}^{2\pi}\ \sin x\ dx$$

• -2

• 1

• $$\frac{2}{\pi}$$

• $$\frac{4}{\pi}$$

Multiple Choice

$$20.\ \int_0^{\frac{\pi}{2}}\ \cos x\ dx$$

• -2

• 1

• $$\frac{2}{\pi}$$

• $$\frac{4}{\pi}$$

Multiple Choice

$$22.\ \int_0^{\frac{\pi}{4}}\ \sec^2x\ dx$$

• -2

• 1

• $$\frac{2}{\pi}$$

• $$\frac{4}{\pi}$$

Finite Integral of Exponential and Logarithmic Functions

Multiple Choice

$$21.\ \int_0^1\ e^x\ dx$$

• -2

• e-1

• $$\frac{4}{\pi}$$

Multiple Choice

$$29.\ \int_1^e\ \frac{1}{x}\ dx$$

• -2

• e-1

• 1