fz² dz along the indicated paths: -1+i -1 C 0 (a) 1 1 + i 17 0 (b) 1

Q: 1 = S I .3 x³ sin(2x) (x² + 1)² -dx.

A:

Q: Find the value of w = 5 bounded by y = 0, y = 2,0 ≤x≤ 2y, and x ≤z≤y + 4.

A: We have to find the value of w=5. It is bounded by:y=0, y=20≤x≤2yx≤z≤y+4

Q: +x-2xy) ds once around the circle x² + y² = 4, z = 0

A: ∮C(2+x−2xy)ds once around the circle x2+y2=4,z=0x2+y2=4Using parametric…

Q: f(t) = {, - 2 2 -2

A:

Q: log (a cos 0+bª sin 6) de=r log ( a+, -), a>0, b>0. 2

A:

Q: 1 Circles

A: To Setup: Integrals for the area of the shaded region in both figures.

Q: Find the volume of a sphere. Use cylindrical coordinates. Find the volume of a circular cone.

A:

Q: /6х — 22

A: We will evaluate the given integral as shown below by using trigonometric substitution

Q: (1 if 0 1

A: The given function is,f(x)=1 if 0<x<10, if x>1

Q: 4x³ (1 - 4x²) 1²/1 2 S- -dx

A:

Q: I = c2π sin² 0 0 5 + 3 cos 0 dᎾ .

A:

Q: Evaluate / (x* – 2y) dA where D = {(x, y) E R²| – 1 < x < 2 and – x² < y < x²}. -

A:

Q: S(1) = >1

A:

Q: 3ry dy dr, V16 z 3

A:

Q: Given the line integral | (2x2° + 6y) da + (6x – 2yz) dy + (3a² z² – y²) dz. Let C be the path that…

A: Note that the following integral reduces like this :

Q: Use Green's Theorem to evaluate the line integral (y – x)dx + (2x – y)dy for the given path. C: x =…

A: Given integral is ∫Cy−xdx+2x−ydy where C:x=2cosθ,y=sinθ,0≤θ≤2π. Green theorem is given by…

Q: 2ху dx + (x + у) dy Jc C: boundary of the region lying between the graphs of y = 0 and y = 1 – x2

A: Given: ∫C2xy dx+(x+y)dy y=0 and y=1-x2

Q: Can

A: We use Cauchy's Residue Theorem

Q: r2 √4-x² To √√₁-(x-1)² x √√√x² + y² dydx

A:

Q: Evaluate 9 €₂ N d da (2-1) (2³8) ²

A: Using,  Cauchy integral theorem we can solve above integral.

Q: 21.3 xdx V16 - 9х4 0.

A: We to integrate the given integration.

Q: 2. CG X It otx

A:

Q: if 01

A:

Q: Evaluate the definite integral 2x dx in two -3 different ways: (2a) as a net area. Sketch the…

A: From the above graph, the net area is area from 0 to 1 = 1 area from -3 to 0 is -9. So net area =…

Q: Explain how the fundamental theorem of calculus relates integrals and derivatives. Formats T B T U…

A: Fundamental theorem of calculus relates integrals and derivatives.

Q: 5. cot x cos²x dx In

A: We will evaluate the given integral as shown below

Q: Explain Integrals of Vector Functions?

A: A function of the form rt=ft i^+gt j^+ht k^  or rt=ft, gt, ht is called a vector-valued function in…

Q: What is the value of the contour integral of a complex function over a closed curve?

A: The value of the contour integral of a complex function over a closed curve is zero if the function…