We can define an improper integral over the plane R² as a limit: [[[..€¯ e-x²-y² dxdy = = JSD. € lim a✈00 e-²-² dxdy where Da is the disk around 0 of radius a. Use this definition to prove [[[₁₂₁ ₁²² dady = π R² that

Q: (b) Use the Fundamental Theorem of Calculus to evaluate the integral .8 So da dx x-6 x+8

A: Need to use the fundamental theorem of calculus to solve this integral.

Q: 5. Is there a potential F(x, y) for f(x, y) = (y² +3x²)i+2xyj? If so, find one.

A: The given function is Let the potential function is Then

Q: Evaluate the following integral. Assume a and b are real numbers and n is an integer. Sx(ax + b) dx;…

A: By considering u=ax+b we can integrate it.

Q: -1 tanx 4. 1+x?

A:

Q: xp - )s ɛ

A: As per the guidelines I can solve only first question

Q: Use a trigonometric substitution to evaluate the following integral. (36+ x²)-3/2dx

A:

Q: Show all definite integrals. Find the area of the surface generated by revolving the curve y=3x-x2…

A:

Q: dx (§5.7, Q28) Evaluate the indefinite integral: (x – 9)5"

A: step 2 complete solution

Q: Evaluate the integral, if it exists: +z sinx dx z-

A: We will evaluate the integral if it exists in the following step.

Q: Evaluate the following indefinite integrals: 1+e (³) ſ a) Sites de (b) / 2 [2√1-2² dz. Justify your…

A: I am going to solve the problem by using some simple calculus to get the required result of the…

Q: Evaluate the following integrals using theFundamental Theorem of Calculus.

A: By the fundamental theorem of calculus, we get ∫abf'(x)dx = f(b)  - f(a) Here f'(x) = 4x3 Integrate…