# 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 = kx^{n} where n ≥ 0. What is the location of its centroid from the line x = b? Determine also the y coordinate of the centroid.

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## 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 y^{2} = kx.

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## 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.

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## 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

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

**Example 6**

What is the area within the curve r^{2} = 16 cos θ?

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

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

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

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## 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*.

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## Example 6 | Plane Areas in Rectangular Coordinates

**Example 6**

Find each of the two areas bounded by the curves *y* = *x*^{3} - 4*x* and *y* = *x*^{2} + 2*x*.

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## Example 5 | Plane Areas in Rectangular Coordinates

**Example 5**

Find the area between the curves 2*x*^{2} + 4*x* + *y* = 0 and *y* = 2*x*.

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