# Vertical Strip

## Area, moment of inertia, and radius of gyration of parabolic section

**Situation**

Given the parabola 3x^{2} + 40y – 4800 = 0.

Part 1: What is the area bounded by the parabola and the X-axis?

A. 6 200 unit^{2}

B. 8 300 unit^{2}

C. 5 600 unit^{2}

D. 6 400 unit^{2}

Part 2: What is the moment of inertia, about the X-axis, of the area bounded by the parabola and the X-axis?

A. 15 045 000 unit^{4}

B. 18 362 000 unit^{4}

C. 11 100 000 unit^{4}

D. 21 065 000 unit^{4}

Part 3: What is the radius of gyration, about the X-axis, of the area bounded by the parabola and the X-axis?

A. 57.4 units

B. 63.5 units

C. 47.5 units

D. 75.6 units

## 709 Centroid of the area bounded by one arc of sine curve and the x-axis

**Problem 709**

Locate the centroid of the area bounded by the x-axis and the sine curve $y = a \sin \dfrac{\pi x}{L}$ from x = 0 to x = L.

## 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.

## 707 Centroid of quarter ellipse by integration

**Problem 707**

Determine the centroid of the quadrant of the ellipse shown in Fig. P-707. The equation of the ellipse is $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$.

## 706 Centroid of quarter circle by integration

**Problem 706**

Determine the centroid of the quarter circle shown in Fig. P-706 whose radius is r.

## 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.

## 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*.

## 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*.

## Example 4 | Plane Areas in Rectangular Coordinates

**Example 4**

Solve the area bounded by the curve *y* = 4*x* - *x*^{2} and the lines *x* = -2 and *y* = 4.

## Example 2 | Plane Areas in Rectangular Coordinates

**Example 2**

Find the area bounded by the curve *a*^{2}*y* = *x*^{3}, the *x*-axis and the line *x* = 2*a*.