find the area of the region bounded by the parabola y^2= 16x and its latus rectum
64/3 sq. units
To find the area of the region bounded by the parabola y^2 = 16x and its latus rectum, we need to determine the vertices of the parabola and the length of the latus rectum.
1. The equation of the parabola is y^2 = 16x. We can rewrite this equation as x = (1/16) * y^2. This equation represents a parabola that opens to the right.
2. To find the coordinates of the vertex, we let y = 0 and solve the equation. Substituting y = 0 into x = (1/16) * y^2, we get x = (1/16) * 0^2 = 0. Therefore, the vertex of the parabola is (0, 0).
3. Next, we need to find the length of the latus rectum. The latus rectum of a parabola with equation x = (1/4a) * y^2 is given by the formula 4a, where 'a' is the coefficient of y^2. In this case, 'a' is (1/16), so the length of the latus rectum is 4 * (1/16) = 1/4.
4. The latus rectum is a line segment perpendicular to the axis of symmetry, passing through the vertex. Since the latus rectum is perpendicular to the x-axis, it has an equation of the form x = h, where h is the x-coordinate of the vertex. Therefore, the equation of the latus rectum is x = 0.
5. Now, we have the vertex (0, 0) and the equation of the latus rectum x = 0. We can visualize that the region bounded by the parabola and the latus rectum is a quarter of the parabolic shape. The area of this region is given by the formula 1/4 * (length of latus rectum)^2.
6. Plugging in the values, we have 1/4 * (1/4)^2 = 1/16 square units.
Therefore, the area of the region bounded by the parabola y^2 = 16x and its latus rectum is 1/16 square units.
To find the area of the region bounded by the parabola y^2 = 16x and its latus rectum, we need to first understand the properties of a parabola and its latus rectum.
A parabola is a U-shaped curve that can be represented by the equation y^2 = 4ax, where "a" is a constant. In the given equation, y^2 = 16x, we can see that a = 4.
The latus rectum of a parabola is a line segment perpendicular to the axis of symmetry of the parabola and passes through the focus. The length of the latus rectum is equal to four times the focal length of the parabola. In this case, since the equation of the parabola is y^2 = 16x, we can find the focal length by identifying the value of "a".
In the equation y^2 = 4ax, "a" represents the focal length. Therefore, the focal length in this case is 4.
Now that we know the length of the latus rectum is equal to four times the focal length, we can calculate it by multiplying the focal length by 4: 4 * 4 = 16.
The latus rectum of the parabola intersects the x-axis at a distance of 16 units from the vertex. So, the x-coordinate of the point of intersection is 16.
To find the area bounded by the parabola and the latus rectum, we need to calculate the integral of the function 16x from the point where the parabola and the latus rectum intersect to the vertex of the parabola.
The integral to find the area is given by:
∫(a to b) [f(x)] dx
where f(x) represents the function defining the curve.
In this case, we integrate the function 16x from 16 to 0 (since the vertex of the parabola is located at the origin):
∫(16 to 0) [16x] dx
Integrating this expression gives us the area of the region bounded by the parabola and its latus rectum. Solving this integral will yield the result.