Integral Review

Calculate the definite integral of f(x) = 2x from 0 to 3.

• 9

• 6

• 12

• 5

Find the indefinite integral of g(x) = 3x^2 + 2x - 5.

• (1/3)x^3 + x^2 - 5x

• 3x^2 + 2x + 5

• (1/3)x^3 + x^2 - 5x + C

• (1/3)x^3 + 2x - 5

Explain the concept of area under a curve and how it is related to integrals.

• The concept of area under a curve is related to integrals as integrals are used to calculate the area under a curve by summing up infinitely small areas.

• Area under a curve is determined by counting the number of points on the curve

• Integrals are used to calculate the slope of a curve

• The concept of area under a curve is unrelated to integrals

What are some real-life applications of integrals?

• Determining the boiling point of water

• Measuring the speed of light

• Solving algebraic equations

• Some real-life applications of integrals include calculating areas under curves, determining volumes of irregular shapes, analyzing population growth, and predicting stock prices.

Determine the indefinite integral of k(x) = sin(x) + 2cos(x).

• -cos(x) + sin(x) + C

• -cos(x) + 2sin(x) + C

• -sin(x) + 2cos(x) + C

• -sin(x) + cos(x) + C

How can integrals be used to find the average value of a function over an interval?

• Integrals are used to find the average value of a function over an interval by taking the definite integral of the function over the interval and dividing by the length of the interval.

• Integrals are used to find the maximum value of a function over an interval

• Integrals are used to find the derivative of a function over an interval

• Integrals are used to find the slope of a function over an interval

Discuss the concept of antiderivatives and their relationship to integrals.

• Antiderivatives are the reverse process of differentiation, representing functions whose derivative is a given function.

• Antiderivatives are only used in geometry.

• Integrals are not related to antiderivatives.

• Antiderivatives are the same as derivatives.

1.What should "u" equal in this integral?

• csc(x)

• csc²x

• sin(x)

• cos(x)

6. What should "u" equal in this integral?

• x
• (lnx)/x
• lnx
• sin(lnx)