fine the area bounded by the graphs of y=5x^2, y=20
5 x^2 = 20 --->
x =+/-2
Area is F(2) - F(-2)
where F(x) = 20x - 5/3 x^3
To find the area bounded by the graphs of y=5x^2 and y=20, follow these steps:
1. First, set the two equations equal to each other: 5x^2 = 20.
2. Simplify the equation by dividing both sides by 5: x^2 = 4.
3. Take the square root of both sides to solve for x: x = ±2.
This gives you the x-values for the points where the two graphs intersect.
4. To find the area, you'll need to integrate the function that represents the upper curve (y=20) minus the lower curve (y=5x^2) between the x-values -2 and 2.
The area can be calculated using the formula: Area = ∫(upper curve - lower curve) dx from x = -2 to x = 2.
5. Integrate the function F(x) = 20 - 5x^2 with respect to x from -2 to 2.
F(x) = ∫(20 - 5x^2) dx
6. To integrate F(x), use the power rule of integration.
Integrate term by term: ∫20 dx - ∫5x^2 dx
∫20 dx = 20x
∫5x^2 dx = (5/3) x^3 + C (where C is the constant of integration)
7. Plug in the x-values into the integrated equation and subtract the lower limit from the upper limit to find the area.
Area = F(2) - F(-2) = (20(2) - (5/3)(2)^3) - (20(-2) - (5/3)(-2)^3)
8. Further simplification can be done to find the final numeric value of the area.
Therefore, using these steps, you can find the area bounded by the two given graphs.