If the area under the curve of f(x) = 25 - x^2 from x = -4 to x = 0 is estimated using four approximating rectangles and left endpoints, will the estimate be an underestimate or overestimate?
1) Underestimate
2) Overestimate
3) The area will be exact
GRAPH IT(0at x=-5, 9 at x=-4, 25 at x=0
dy/dx = -2 x
d^2y/dx^2 =-2
This curve curves down (sheds water, d^2y/dx^2 <0 ) while the slope is positive for negative x
Therefore those rectangles do not fill the space under the curve.
In the given region, the graph is moving downwards.
Left endpoint approximation does not cover the entire region, and will hence give an underestimate.
thank you
It is not moving downwards. It is curving downwards :)
To determine whether the estimate of the area under the curve will be an underestimate or an overestimate when using four approximating rectangles and left endpoints, we need to understand how the left endpoint method works.
The left endpoint method, also known as the left Riemann sum, estimates the area under a curve by using rectangles whose left endpoints touch the curve. The width of each rectangle is determined by the partitioning of the interval from x = -4 to x = 0 into four equal sub-intervals.
In this case, the interval is divided into four sub-intervals: [-4, -3], [-3, -2], [-2, -1], and [-1, 0]. The left endpoints of each sub-interval are -4, -3, -2, and -1.
To estimate the area under the curve using the left endpoint method, we calculate the area of each rectangle and sum them up. Since each rectangle's left endpoint touches the curve, the heights of the rectangles are determined by evaluating the function f(x) = 25 - x^2 at those left endpoints.
If we substitute the left endpoints -4, -3, -2, and -1 into the function f(x) = 25 - x^2, we get the following heights for each rectangle:
Rectangle 1: f(-4) = 25 - (-4^2) = 9
Rectangle 2: f(-3) = 25 - (-3^2) = 16
Rectangle 3: f(-2) = 25 - (-2^2) = 21
Rectangle 4: f(-1) = 25 - (-1^2) = 24
Now, we calculate the area of each rectangle by multiplying the height by the width. Since the width of each rectangle is the same, let's assume it to be Δx.
Rectangle 1 area: 9 * Δx
Rectangle 2 area: 16 * Δx
Rectangle 3 area: 21 * Δx
Rectangle 4 area: 24 * Δx
Summing up the areas of all four rectangles, we get:
Total estimated area = (9 * Δx) + (16 * Δx) + (21 * Δx) + (24 * Δx) = (9 + 16 + 21 + 24) * Δx
As we can see, the estimated area is equal to the sum of the heights of the rectangles multiplied by the width Δx. Since the function f(x) = 25 - x^2 is concave down (opening downwards) in the interval [-4, 0], the values of the function decrease as x increases. Hence, the height of each rectangle decreases from left to right.
Therefore, the left endpoint method will underestimate the area under the curve because it only considers the leftmost points of each sub-interval. It leaves out the additional area that would be covered by rectangles with right endpoints and rectangles with higher heights.
In conclusion, the estimate of the area under the curve using four approximating rectangles and left endpoints will be an underestimate.