Find out the answer to "Determine the following definite integrals: ∫−50(1+25−x2)dx"

Brennan Flores

Answered question

2021-09-18

Determine the following definite integrals:

\int_{- 5}^{0} (1 + \sqrt{25 - x^{2}}) d x

Aniqa O'Neill

Skilled2021-09-19Added 100 answers

Evaluate the integrals:

f (x) = \int_{- 5}^{0} (1 + \sqrt{25 - x^{2}}) d x

In terms of areas:

f (x) = \int_{- 5}^{0} \sqrt{25 - x^{2}} d x

Let

x = 5 \sin u

f (x) = {[x]}_{- 5}^{0} + \int_{- \frac{π}{2}}^{0} \sqrt{25 - {(5 \sin u)}^{2}} d u

= {[x]}_{- 5}^{0} + \int_{- \frac{π}{2}}^{0} 25 \cos^{2} (u) d u

= [0 - (- 5)] + 25 \cdot \int_{- \frac{π}{2}}^{0} \cos^{2} (u) d u

= 5 + 25 \cdot \frac{1}{2} \cdot \int_{\frac{π}{2}}^{0} 1 + \cos (2 u) d u

= 5 + \frac{25}{2} [{[u]}_{- \frac{π}{2}}^{0} + {[\frac{1}{2} \sin 2 u]}_{- \frac{π}{2}}^{0}]

= 5 + \frac{25}{2} [[0 - (- \frac{π}{2})] + (0)]

= 5 + \frac{25}{2} [\frac{π}{2} + 0]

= 5 + \frac{25 π}{4}

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