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

Question

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BlackTom AI · Accepted Answer

The integrand is the parabola scaled by 1/7: f(x) = (x^2 − 14x)/7 and the x-axis g(x) = 0. Zeros of f are where x^2 − 14x = 0 ⇒ x(x − 14) = 0, so the curves meet at x = 0 and x = 14. On the interval (0,14) the quadratic x^2 − 14x is negative, so f(x) ≤ 0 there; the area bounded by the curve and the x-axis is the integral of the positive vertical distance |f(x) − 0| = −f(x).

Step-by-step computation
1. Write the area as an integral of the absolute value:
   A = ∫_0^{14} |f(x)| dx = −∫_0^{14} f(x) dx  because f(x) ≤ 0 on (0,14).

2. Substitute f(x) = (x^2 − 14x)/7:
   A = −∫_0^{14} (x^2 − 14x)/7 dx = −(1/7) ∫_0^{14} (x^2 − 14x) dx.

3. Compute the antiderivative:
   ∫(x^2 − 14x) dx = x^3/3 − 7x^2.

4. Evaluate from 0 to 14:
   At x = 14: x^3/3 − 7x^2 = 14^3/3 − 7·14^2 = 2744/3 − 1372.
   Convert 1372 to thirds: 1372 = 4116/3, so 2744/3 − 4116/3 = −1372/3.
   At x = 0 the antiderivative is 0. Thus ∫_0^{14} (x^2 − 14x) dx = −1372/3.

5. Multiply by −(1/7):
   A = −(1/7) · (−1372/3) = (1372) / (21).

6. Simplify 1372/21: divide numerator and denominator by 7 → (1372 ÷ 7)/(21 ÷ 7) = 196/3.

Why this is correct
- The factor 1/7 in f scales the vertical distance by 1/7, so the area is 1/7 of the area you would get for x^2 − 14x above the same interval. The sign flip is handled by the absolute value because the parabola lies below the x-axis between its roots. The arithmetic simplifies cleanly to 196/3.

Why the other choices are incorrect
- Options like 1372/3 would correspond to integrating (x^2 − 14x) without the 1/7 scaling (i.e., missing the factor 1/7).
- Other listed fractions (e.g., 344/3, 229/2, etc.) arise from arithmetic errors or wrong limits; they do not match the correct integral evaluation and simplification above.

Conclusion
Carrying out the definite integral with the correct sign and the 1/7 scaling yields A = 196/3, which matches the provided answer.

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

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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 algebraic functions. 单项选择题

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 equations by integrating (i) with respect to x and (ii) with respect to y.