Asksia AI LOGO

Sia

chen74304's Question
Math
Posted 2 months ago
why
Determine the area under the curve y=4x2y = 4 - x^2 from x=2x = -2 to x=2x = 2.
Sign in to unlock the answer
Answer from Sia
Posted 2 months ago
Solution by Steps
step 1
We need to find the area under the curve y=4x2y = 4 - x^2 from x=2x = -2 to x=2x = 2. This requires evaluating the definite integral 22(4x2)dx\int_{-2}^{2} (4 - x^2) \, dx
step 2
First, we find the antiderivative of the integrand 4x24 - x^2. The antiderivative is 4xx33+C4x - \frac{x^3}{3} + C
step 3
Next, we evaluate this antiderivative at the bounds x=2x = -2 and x=2x = 2
step 4
Evaluating at x=2x = 2: 4(2)(2)33=883=883=24383=1634(2) - \frac{(2)^3}{3} = 8 - \frac{8}{3} = 8 - \frac{8}{3} = \frac{24}{3} - \frac{8}{3} = \frac{16}{3}
step 5
Evaluating at x=2x = -2: 4(2)(2)33=883=8+83=243+83=1634(-2) - \frac{(-2)^3}{3} = -8 - \frac{-8}{3} = -8 + \frac{8}{3} = -\frac{24}{3} + \frac{8}{3} = -\frac{16}{3}
step 6
Subtract the value at x=2x = -2 from the value at x=2x = 2: 163(163)=163+163=323\frac{16}{3} - (-\frac{16}{3}) = \frac{16}{3} + \frac{16}{3} = \frac{32}{3}
step 7
Therefore, the area under the curve y=4x2y = 4 - x^2 from x=2x = -2 to x=2x = 2 is 32310.667\frac{32}{3} \approx 10.667
Answer
The area under the curve is 32310.667\frac{32}{3} \approx 10.667.
Key Concept
Definite Integral
Explanation
The definite integral of a function over an interval gives the area under the curve of that function over that interval.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages
Strong algorithms that better know you
Early access to new release features
Study Other Question