Find the area bounded by the parabola y^2=4x and the line y=2x-4. Use vertical representative rectangles (integrate with respect to x) and horizontal representative rectangles (integrate with respect to y). the answer is 9 square units ... i just need to know how to get to that.
You may find it simpler to rotate the curve and the line a quarter turn anti-clockwise around the origin.
This gives y = x^2/4 and y = 1/2x + 2
They intersect at x = -2 and 4.
Find the definite integral over the interval (-2, 4) of 1/2x + 2 - x^2/4.
This gives 9 units
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ur dumb ok???
To find the area bounded by the parabola y^2 = 4x and the line y = 2x - 4 using vertical representative rectangles (integrating with respect to x) and horizontal representative rectangles (integrating with respect to y), you can follow these steps:
1. Vertical Representative Rectangles (Integration with respect to x):
- First, visualize the region bounded by the parabola y^2 = 4x and the line y = 2x - 4.
- To find the x-values where the parabola and the line intersect, set them equal to each other:
4x = (2x - 4)^2
- Simplify the equation and solve for x. You should find two x-values.
- Identify the region between these two x-values.
- Integrate the function y = (2x - 4) - sqrt(4x) with respect to x over this region.
- Evaluate the integral and find the area.
2. Horizontal Representative Rectangles (Integration with respect to y):
- Rearrange the equation y^2 = 4x to obtain x in terms of y: x = y^2/4.
- Find the y-values where the parabola and the line intersect by setting y = 2x - 4 equal to y^2/4:
y = (y^2/4 - 4)
- Simplify the equation and solve for y. You should find two y-values.
- Identify the region between these two y-values.
- Integrate the function x = y^2/4 - (2x - 4) with respect to y over this region.
- Evaluate the integral and find the area.
By using either the vertical representative rectangles or the horizontal representative rectangles method, you should arrive at the same answer of 9 square units.