How do I evaluate the integral: #int_0^(7sqrt(3/2))dx / sqrt(49-x^2)#?

Answer 1

First, recall the antiderivative formula:
#int dx/sqrt(a^2-u^2) = arcsin(u/a) + C#

Thus, #int dx/sqrt(49-x^2) = int dx/sqrt(7^2-x^2)#

We have #int dx/sqrt(7^2-x^2) = arcsin(x/7) + C#

Substitute in the constraints we

#int_0^(7sqrt(3/2)) dx/sqrt(7^2-x^2) = arcsin((7sqrt(3/2))/7) - arcsin(0) = arcsin((7sqrt(3/2))/7) #

Please find the formula sheet below for reference:

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To evaluate the integral ( \int_0^{7\sqrt{\frac{3}{2}}} \frac{dx}{\sqrt{49-x^2}} ), we can use trigonometric substitution. Let ( x = 7\sin(\theta) ), then ( dx = 7\cos(\theta)d\theta ).

Substituting these into the integral, we get:

[ \int_0^{\frac{\pi}{2}} \frac{7\cos(\theta)d\theta}{\sqrt{49 - 49\sin^2(\theta)}} ]

[ = \int_0^{\frac{\pi}{2}} \frac{7\cos(\theta)d\theta}{\sqrt{49\cos^2(\theta)}} ]

[ = \int_0^{\frac{\pi}{2}} \frac{7\cos(\theta)d\theta}{7\cos(\theta)} ]

[ = \int_0^{\frac{\pi}{2}} d\theta ]

[ = \left[\theta\right]_0^{\frac{\pi}{2}} ]

[ = \frac{\pi}{2} ]

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7