Find the shaded area above.

Question

BlackTom AI · Accepted Answer

Area between curves y = 2x + 1 (top) and y = x (bottom) from x = 0 to x = 2 is given by the integral of (top − bottom) dx.

1) Relevant knowledge points and concepts
- The area between two curves y = f(x) (upper) and y = g(x) (lower) on [a,b] is ∫[a to b] (f(x) − g(x)) dx.
- Identify vertical boundaries a and b from the picture: the shaded region is bounded on the left by x = 0 (the y‑axis) and on the right by x = 2 (the dashed vertical line).

2) Complete reasoning process step by step
- Determine which curve is on top on [0,2]: for any x, 2x+1 − x = x+1, which is positive on [0,2], so y = 2x + 1 is above y = x.
- Set up the area integral from x = 0 to x = 2 of (2x + 1 − x) dx = ∫[0 to 2] (x + 1) dx.

3) Calculations with intermediate steps
- Integrand: x + 1.
- Antiderivative: (1/2)x^2 + x.
- Evaluate from 0 to 2:
  (1/2)(2)^2 + (2) − [ (1/2)(0)^2 + 0 ] = (1/2)(4) + 2 = 2 + 2 = 4.

4) Why the correct answer is correct
- The integral computes the exact vertical distance between the two lines at each x, summed over 0 ≤ x ≤ 2. The computation yields 4 square units, so the shaded area is 4.

5) Why the other options are incorrect
- (a) 2 sq. units: too small — corresponds roughly to half the true integral; incorrect.
- (c) 16 sq. units: far too large — would be area of a much larger region.
- (d) 8 sq. units: double the correct value; not supported by the computed integral.

6) Conclusion
The area of the shaded region is 4 square units, matching the provided correct answer: b. 4 sq. units.

Find the shaded area above.单项选择题

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Find the shaded area above.单项选择题

类似问题

Which one of the following is equal to the shaded area above?

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Which one of the following is equal to the shaded area above?

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Find the area of the region bounded by the graphs of the function , , . Round your answer to three decimal places. ​

Find the area of the region bounded by the graphs of the algebraic functions. ​ ​

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Question at position 8 The exact area of the region bounded by the graphs of y = x and y = x2 is1/3 sq unit.1/6 sq unit.1/2 sq unit.2/3 sq unit.5/6 sq unit.

更多留学生实用工具

希望你的学习变得更简单

Find the area of the region bounded by the graphs of the function , , . Round your answer to three decimal places.

Find the area of the region bounded by the graphs of the algebraic functions.

Find the area of the region bounded by the equations by integrating (i) with respect to x and (ii) with respect to y.