Area of the Region and Volume of the Solid of Revolution Given: The region is bounded by the curves y = x² - 1, y = 1, x = 0, and x = 1. a. Draw a sketch of the region. b. Find the area of the region.

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A: Answer is following.

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A: The integration of the integral is |2x+1| limit from -2 to 2.

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A: Solution

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A: Solve the differential equation: (1+x3)dy - x2ydx=0; y(1)=2

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A: Graph

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A: Please see the below picture for detailed solution.

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A: Topic:- application of integration

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A: see below the answer and calculation

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Q: 2. Given f(x, y) = 12x2x³ + 3y² + 6xy. (i) Find critical points of f. (ii) Use the second derivative…

A: To find out the critical points of f.

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A: Solution:

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A: We have to determine integration

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A: Here, x6cosx3 is an even function. So, ∫-11 x6cosx3dx≠0

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A: solution is given below

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A: Let's solve for area and volume of given two curves.

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A: Let's solve.

Q: 7. fcsc (3x) • cot (3x) dx csc(3x) A. + C B. -csc(3x) 3 + C C. -3 csc(3x) + C

A: As per guidelines we can answer only one question at a time, so if you need more answers kindly…

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Q: Question 13 x₁ - 3x3 = -2 Find the solution to the system of equation 3x₁+x₂-2x3=5₁ 2x₁+2x₂+x3=4 A)…

A: By Gauss Jordan elimination

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Q: Show

A: Please refer to the image below

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A: We can find the integration as below.

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A: Topic = Limits

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A: Given,  ∫012x+22dx ,  n=4

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A: Given