Sovi AI

Sovi AI

Assistant de questions photo
ENFRDEESJAZH-CNZH-TW

evaluate $int_{-8}^{0} (-sqrt{-x^{2}-8x})dx$

$-8\pi$

Catégorie: calculus Mis à jour: 2026-02-09

Question

evaluate $int_{-8}^{0} (-sqrt{-x^{2}-8x})dx$

Étapes de solution

  1. Understand the question

    evaluate $int_{-8}^{0} (-sqrt{-x^{2}-8x})dx$

  2. 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:
    $$-8\pi$$

  3. Final answer

    $-8\pi$

Réponse

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 enlarge

Question 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 $int_{-8}^{0} (-sqrt{-x^{2}-8x})dx$

Sujets liés

mathematicscalculuscalculationhigh schoolturns-1

Questions liées

Sovi AI iOS

Site officiel : mysovi.ai. Les pages de questions sont servies sur question-banks.mysovi.ai. L’app iOS est disponible sur l’App Store Apple.

Télécharger sur l'App Store Catégorie: Calcul