Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.
y = x^−3, 1 ≤ x ≤ 5
To give a rough estimate of the area beneath the curve, we can use a graph.
1. First, plot the curve y = x^(-3) on a graph with the x-axis ranging from 1 to 5.
2. Divide the area beneath the curve into rectangles of equal width. The narrower the width of the rectangles, the more accurate our estimate will be.
3. Draw rectangles whose height is determined by the curve. For each rectangle, select a representative point on the curve (e.g., the left endpoint, the right endpoint, or the midpoint) and draw a rectangle with that height.
4. Add up the areas of all the rectangles. The sum of their areas will give us a rough estimate of the area beneath the curve.
To find the exact area beneath the curve, we need to evaluate the definite integral of the function y = x^(-3) over the interval [1, 5].
1. Calculate the antiderivative (or indefinite integral) of the function y = x^(-3).
Integrate: ∫ x^(-3) dx
The antiderivative of x^(-3) is (-1/2) * x^(-2), so the indefinite integral is (-1/2) * x^(-2) + C, where C is the constant of integration.
2. Evaluate the definite integral using the fundamental theorem of calculus.
Use the evaluated antiderivative: ∫[1, 5] x^(-3) dx = [(-1/2) * x^(-2)] evaluated from x = 1 to x = 5.
Plug in the upper limit (5) and the lower limit (1) into the antiderivative expression and subtract the lower limit from the upper limit:
[(-1/2) * 5^(-2)] - [(-1/2) * 1^(-2)]
Simplify the expression:
[(-1/2) * 1/25] - [(-1/2) * 1]
-(1/50) + (1/2) = (1/2) - (1/50)
Therefore, the exact area beneath the curve is (1/2) - (1/50), or 23/50.