9. Graphical Applications of Integrals

Find the area of the region described in the following exercises.

The region bounded by y=|x−3|and y=x/2

Center of gravity: Find the center of gravity of the region bounded by the x-axis, the curve y = sec x, and the lines x = -pi/4 and x = pi/4.

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.

x = 2y², x = 0, y = 3

Find the area of the shaded region ONLY that lies between the line $y=1$ & $f\left(x\right)=\sin 2x$.

Find the area of the region described in the following exercises.

The region bounded by y=2 / 1 + x^2 and y=1

21–30. {Use of Tech} Arc length by calculator

b. If necessary, use technology to evaluate or approximate the integral.

y = cos 2x, for 0 ≤ x ≤ π

58–61. Arc length Find the length of the following curves.

y = x³/6 + 1/2x on [1,2]

Determine the area of the shaded region bounded by the curve x^2=y^4(1−y^3) (see figure).

Determine the area of the shaded region in the following figures.

(Hint: Find the intersection point by inspection.)

Find the area of the region described in the following exercises.

The region bounded by y=√x, y=2x−15, and y=0

Find the extreme values of ƒ(x) = x³ - 3x², and find the area of the region enclosed by the graph of ƒ and the x-axis.

Find the area of the shaded region between $f\left(x\right)=x^4-x^2$ & $g\left(x\right)=3x^2$.

20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.

b. Find the centroid of the region.

Centroid:

Find the centroid of the region cut from the first quadrant by the curve

y = 1/√(x + 1) and the line x = 3.

For the given regions R₁ and R₂, complete the following steps.

a. Find the area of region R₁.

R₁ is the region in the first quadrant bounded by the y-axis and the curves y=2x^2 and y=3−x; R₂ is the region in the first quadrant bounded by the x-axis and the curves y=2x^2 and y=3−x(see figure).