Use a definite integral to find an expression that

Question

Evaluate the integral.

1) ?⁵₀(2x² + x + 2) dx

Use a definite integral to find an expression that represents the area of the region between the given curve and the x-axis on the interval[0, b].

2) y=12??x²

Evaluate the integral.

3) ? x⁴ (x⁵ - 8)⁴dx

Solve the differential equation.

4) dy/dx = 9 √xy

Use the definite integral to find the area between the x-axis and f(x) over the indicated interval.

5) f(x)= x²- 6x+ 9;  [2, 4]

Solve the initial value problem.

6) dy/dx + xy = 2x;y(0) = 5

Find the area between the curves.

7) y= x,  y= x²

Evaluate the integral.

8) ?23x sin x dx

9) ?⁴₂ 6x In x dx

10) ? dx/(16 - x²)²

Evaluate the improper integral or state that it is divergent.

11) ?⁰_-x 16 e^x sin x dx

Find the area or volume.

12) Find the volume of the solid generated by revolving the region under the curvey=6e-^x2 in the first quadrant about the y-axis.

Decide whether or not the function is a probability density function on the indicated interval.

13) f(x) = 4/74 x³; (0, 3)

Find the expected value of the probability density function to the nearest hundredth.

14) f(x) = 3x-4; (1, ∞)