Area by Integration

Problem 708
Compute the area of the spandrel in Fig. P-708 bounded by the x-axis, the line x = b, and the curve y = kxⁿ where n ≥ 0. What is the location of its centroid from the line x = b? Determine also the y coordinate of the centroid.
 

Problem 705
Determine the centroid of the shaded area shown in Fig. P-705, which is bounded by the x-axis, the line x = a and the parabola y² = kx.
 

Problem
The figure shown below is composed of arc of circles with centers at each corner of the square 20 cm by 20 cm. Find the area inside the square but outside the region commonly bounded by the quarter circles. The required area is shaded as shown in the figure below.
 

The total surface area of the sphere is four times the area of great circle. To know more about great circle, see properties of a sphere. Given the radius r of the sphere, the total surface area is
 

Example 6
What is the area within the curve r² = 16 cos θ?
 

Find the area enclosed by four-leaved rose r = a cos 2θ.
 

Example 3
Find the area inside the cardioid r = a(1 + cos θ) but outside the circle r = a.
 

Example 6
Find each of the two areas bounded by the curves y = x³ - 4x and y = x² + 2x.
 

Example 5
Find the area between the curves 2x² + 4x + y = 0 and y = 2x.