evaluate $int_{-8}^{0} (-sqrt{-x^{2}-8x})dx$
$-8\pi$
题目
evaluate $int_{-8}^{0} (-sqrt{-x^{2}-8x})dx$
解题步骤
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Understand the question
evaluate $int_{-8}^{0} (-sqrt{-x^{2}-8x})dx$
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Explanation
Step1: Complete the square inside root
Rewrite the quadratic expression under the square root by completing the square:
$$-x^2 -8x = -(x^2 +8x) = -(x^2+8x+16-16) = 16-(x+4)^2$$
The integral becomes:
$$\int_{-8}^{0} -\sqrt{16-(x+4)^2} dx$$Step2: Use substitution $u=x+4$
Let $u = x+4$, so $du = dx$. When $x=-8$, $u=-4$; when $x=0$, $u=4$. The integral transforms to:
$$-\int_{-4}^{4} \sqrt{16-u^2} du$$Step3: Interpret integral as semicircle area
The integral $\int_{-4}^{4} \sqrt{16-u^2} du$ represents the area of a semicircle with radius $4$ (since $y=\sqrt{16-u^2}$ is the upper half of the circle $u^2+y^2=16$). The area of a full circle is $\pi r^2 = 16\pi$, so the semicircle area is $\frac{1}{2} \times 16\pi = 8\pi$.
Step4: Apply the negative sign
Multiply by the negative sign from the integral:
$$-8\pi$$ -
Final answer
$-8\pi$
答案
Explanation
Step1: Complete the square inside root
Rewrite the quadratic expression under the square root by completing the square:
$$-x^2 -8x = -(x^2 +8x) = -(x^2+8x+16-16) = 16-(x+4)^2$$
The integral becomes:
$$\int_{-8}^{0} -\sqrt{16-(x+4)^2} dx$$
Step2: Use substitution $u=x+4$
Let $u = x+4$, so $du = dx$. When $x=-8$, $u=-4$; when $x=0$, $u=4$. The integral transforms to:
$$-\int_{-4}^{4} \sqrt{16-u^2} du$$
Step3: Interpret integral as semicircle area
The integral $\int_{-4}^{4} \sqrt{16-u^2} du$ represents the area of a semicircle with radius $4$ (since $y=\sqrt{16-u^2}$ is the upper half of the circle $u^2+y^2=16$). The area of a full circle is $\pi r^2 = 16\pi$, so the semicircle area is $\frac{1}{2} \times 16\pi = 8\pi$.
Step4: Apply the negative sign
Multiply by the negative sign from the integral:
$$-8\pi$$
Answer
$-8\pi$
Question Image
Click to enlargeQuestion Analysis
Subject
mathematics
Sub Subject
calculus
Education Level
high school
Difficulty
unspecified
Question Type
calculation
Multi Question
No
Question Count
1
Analysis Status
completed
Analyzed At
2026-02-09T19:59:03
OCR Text
Show OCR extraction
evaluate $int_{-8}^{0} (-sqrt{-x^{2}-8x})dx$
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