Find the area of the shaded region.

Question

BlackTom AI · Accepted Answer

The problem asks for the area of the shaded region under the curve y = √x − 3 between x = 0 and x = 16, with the shading representing the area both below and above the x-axis where the curve crosses y = 0.
First, identify where the curve crosses the x-axis: set √x − 3 = 0, so √x = 3 and x = 9. This splits the interval [0, 16] into two parts: from 0 to 9 the curve lies below the x-axis (y ≤ 0), and from 9 to 16 the curve lies above the x-axis (y ≥ 0).
Now compute the area as the sum of the absolute areas on each subinterval:
- On [0, 9], the area between the x-axis and the curve is ∫0^9 (3 − √x) dx. Evaluating, ∫0^9 3 dx = 27 and ∫0^9 √x dx = (2/3)x^(3/2)|0^9 = (2/3)·27 = 18. Therefore, area = 27 − 18 = 9.
- On [9, 16], the area between the curve and the x-axis is ∫9^16 (√x − 3) dx. Evaluating, ∫9^16 √x dx = (2/3)x^(3/2)|9^16 = (2/3)(64 − 27) = (2/3)(37) = 74/3, and ∫9^16 3 dx = 3(x)|9^16 = 3(16 − 9) = 21. Therefore, area = 74/3 − 21 = 74/3 − 63/3 = 11/3.
Add the two subareas: 9 + 11/3 = 27/3 + 11/3 = 38/3.
Now compare with the given options:
- 38/3: matches the computed total area.
- 22/3: would correspond to an under- or over-estimate that omits part of the area (not matching the split integral).
- 29/3: also does not equal the sum of the two subareas derived above.
- 16/3: equals the absolute area if only the portion from 0 to 16 were integrated without splitting at the x-axis, which is not correct for this shaded region since part lies below the axis and must be accounted for separately.
Thus, the correct area is 38/3. The other choices arise from incorrectly handling the split at the x-axis or from summing only part of the region.

MATH2415W16 Quiz 4 Dec 16 (W16)- Requires Respondus LockDown Browser

Find the area of the shaded region.

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MATH*2415*W16 Quiz 4 Dec 16 (W16)- Requires Respondus LockDown Browser

Find the area of the shaded region.

查看解析

登录即可查看完整答案

类似问题

Find the signed area produced by:[math: A=∫364cos(5x)dx](square units)Calculate your answer correct to 4 decimal places.(Note: Make sure your calculator is in radian mode.)

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希望你的学习变得更简单

MATH2415W16 Quiz 4 Dec 16 (W16)- Requires Respondus LockDown Browser