Area by Integration

708 Centroid and area of spandrel 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.
 

Centroid and area of spandrel under the curve y = kx^n

 

705 Centroid of parabolic segment by integration

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.
 

Open to the right parabola in the first quadrant

 

Example 7 | Area inside the square not common to the quarter circles

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.
 

Intersection of circular quadrants

 

Derivation of Formula for Total Surface Area of the Sphere by Integration

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
 

06 Area Within the Curve r^2 = 16 cos θ

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

05 Area Enclosed by Four-Leaved Rose r = a cos 2θ

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

04 Area of the Inner Loop of the Limacon r = a(1 + 2 cos θ)

Example 4
Find the area of the inner loop of the limacon r = a(1 + 2 cos θ).
 

03 Area Inside the Cardioid r = a(1 + cos θ) but Outside the Circle r = a

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

Example 6 | Plane Areas in Rectangular Coordinates

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

Example 5 | Plane Areas in Rectangular Coordinates

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

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