Find out the answer to "Evaluate the integral. ∫024−x2dx"

krypsojx

Answered question

2021-11-16

Evaluate the integral.

\int_{0}^{2} \sqrt{4 - x^{2}} d x

Susan Yang

Beginner2021-11-17Added 20 answers

Step 1
To evaluate:

\int_{0}^{2} \sqrt{4 - x^{2}} d x

.
Solution:
We know that,

\int \sqrt{a^{2} - x^{2}} d x = \frac{1}{2} x \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \sin^{- 1} \frac{x}{a} + C

Using the above formula, we will evaluate the integral.

\int_{0}^{2} \sqrt{4 - x^{2}} d x = \int_{0}^{2} \sqrt{2^{2} - x^{2}} d x

= {[\frac{1}{2} x \sqrt{2^{2} - x^{2}} + \frac{2^{2}}{2} \sin^{- 1} (\frac{x}{2})]}_{0}^{2}

= [(\frac{1}{2} \cdot 2 \cdot \sqrt{2^{2} - 2^{2} + 2 \sin^{- 1} (\frac{2}{2})}) - (0 + 2 \sin^{- 1} 0)]

= [(0 + 2 \sin^{- 1} 1) - 0]

2 \cdot \frac{π}{2}

= π

Step 2
Hence, required answer is

π

Clis1955

Beginner2021-11-18Added 9 answers

Step 1: If f(x) is a continuous function from a to b, and if F(x) is its integral, then:

\int_{a}^{b} f (x) d x = F (x) ∣_{a}^{b} = F (b) - F (a)

Step 2: In this case,

f (x) = \sqrt{4 - x^{2}}

. Find its integral.

4 (\frac{\sin^{- 1} (\frac{x}{2})}{2} + \frac{x \sqrt{1 - \frac{x^{2}}{4}}}{4}) ∣_{0}^{2}

Step 3: Since

F (x) ∣_{a}^{b} = F (b) - F (a)

, expand the above into F(2)−F(0):

4 (\frac{\sin^{- 1} (\frac{2}{2})}{2} + \frac{2 \sqrt{1 - \frac{2^{2}}{4}}}{4}) - 4 (\frac{\sin^{- 1} (\frac{0}{2})}{2} + \frac{0 \sqrt{1 - \frac{0^{2}}{4}}}{4})

Step 4: Simplify.

π

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