Estimate the area under the curve f(x)=16-x^2 from x=0 to x=3 by using three inscribed (under the curve) rectangles. Answer to the nearest integer.
just did this quiz and the answer is 34. 39 is wrong because that is the actual area, and the question says inscribed, which is an underestimate of the actual area. if it asked for circumscribed the answer would be an overestimate of the actual area (39) and would be 43.
again with the Riemann sums?
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To estimate the area under the curve, we can use the method of inscribed rectangles. This involves dividing the x-interval into equal subintervals and approximating the area under the curve within each interval using rectangles that are inscribed (i.e., fit under) the curve.
In this case, we need to estimate the area under the curve f(x) = 16 - x^2 from x = 0 to x = 3 using three inscribed rectangles.
First, we need to determine the width of each rectangle. Since we are dividing the interval from x = 0 to x = 3 into three equal subintervals, each rectangle will have a width of (3 - 0) / 3 = 1.
Next, we need to calculate the height of each rectangle. For inscribed rectangles, we evaluate the function at the left endpoint of each subinterval to determine the height.
Let's evaluate the function at the left endpoint of each subinterval:
- For the first subinterval: x = 0. Plugging this value into f(x), we get f(0) = 16 - 0^2 = 16.
- For the second subinterval: x = 1. Plugging this value into f(x), we get f(1) = 16 - 1^2 = 15.
- For the third subinterval: x = 2. Plugging this value into f(x), we get f(2) = 16 - 2^2 = 12.
Now, we can calculate the area of each rectangle by multiplying the width and height:
- Area of the first rectangle = 1 * 16 = 16 square units.
- Area of the second rectangle = 1 * 15 = 15 square units.
- Area of the third rectangle = 1 * 12 = 12 square units.
To estimate the total area under the curve, we sum up the areas of the three rectangles:
Total estimated area = 16 + 15 + 12 = 43 square units.
Therefore, the estimated area under the curve f(x) = 16 - x^2 from x = 0 to x = 3, using three inscribed rectangles, is approximately 43 square units. Answered to the nearest integer.
so the width of each of the 3 rectangles from x = 0 to 3 is 1 unit
the height, when x = 1 is 15
the height, when x = 2 is 12
the height, when x = 3 is 7
so the estimated area = 1x15 + 1x12 + 1x7 = 34 square units
real answer:
∫ 16-x^2 dx from 0 to 3
= [16x - x^3 /3] from 0 to 3
= 48 - 9 - 0 = 39