Exercise 9: Let R>r > 0. Suppose we have a circle of radius R and a circle of radius r, such that the distance between the centers of the circles is vR² – r2. Find the area lying within the smaller circle, but outside of the larger circle. Hint: Draw the picture.

Q: Consider two concentric cirdes with center O. Here AB, CD & EF are three chords of the outer circle,…

A: Solution is here

Q: Evaluate y*dz – r*dy, where C is the boundary of the region between the circles r? + y? = 4 and z² +…

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Q: Find the volume of the 'ice-cream' that is bounded by x2 + y? + z2 99 and the upper cone z? = 3x2 +…

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Q: Let B be the solid whose base is the circle x^2+y^2=49 and whose vertical cross sections…

A: Given, Circle- x²+y²=49=7² --------equation 1 The equation is of the form x²+y²=r². Therefore, the…

Q: Evaluate , ydV where E is the hemispherical solid region x? + y? + z? 0.

A: x^2+y^2+z^2 <=64, y>=0

Q: Calculate the surface area of the following: the cap cut from the sphere r² + y² + z² = 3 by the…

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Q: Let B be the solid whose base is the circle (x^2) + (y^2) = (4^2)  and whose vertical cross sections…

A: This question is based on application of Integration.

Q: Find the minimum distance from the surface to the given point. (Hint: Start by minimizing the square…

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Q: |Find the average distance from the origin over all the points in the unit circular disk 2² +y? <1.…

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Q: dr Let A be the area of a circle with radiusr. If dt dA = 5, find when r = 5. dt %3D

A: Here we use the application of derivative.

Q: Let A be the area of a circle with radius r. If = 2, find when r = 3. di dt

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Q: Assume a to be a nonzero real no.. Determine the area of the portion of surface xy = az in the fir…

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Q: Find the real number b such that the volume of solid B (r, +

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Q: Use Green's theorem to evaluate O -x²ydx + x°dy, where C is the circle x? + y? = 4 with a…

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Q: Use Green's Theorem to evaluate the integral (3x² – y)dx + 3xdy where C is the circle a? + y? = 1…

A: As we know from the Green's theorem; ∫CMdx+Ndy=∫∫D∂N∂x-∂M∂ydA   Given; I=∫C3x2-ydx+3xdy So, M=3x2-y…

Q: dr Let A be the area of a circle with radius r. If dt 4, find dA when r = 3. dt

A: We are going to solve

Q: Calculate the area of the surface S. x2 that lies between z = 2 and z 6. S is the portion of the…

A: A coordinate system is a convention that is used to represent a point uniquely using a set of…

Q: 2. Find the area of the part of the surface z = cylinder x2 + y² = 9. 3xy that lies within the

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Q: Find the surface area of a sphere with radius, r=8 in. S.A. = in (Type an exact answer in terms of…

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Q: The area of the surface p = 2, 0sz<5,0<¢< T is 201 Select one: O True O False

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Q: Evaluate y*dx – a*dy, where C is the boundary of the region between the circles a? + y² = 16 and %3D…

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Q: Find the Volume r= 2cos 0 , x2+yナ ?= 4.

A: Topic:- volume

Q: (d) Consider the parallelepiped P in R³ whose adjacent sides are (0, 4, 0), (3, 0, 0) and (-1, 1,…

A: Since the student has posted multiple questions and does not mention any specific question, so we…

Q: Evaluate O y°dx – x°dy, where C is the boundary of the region between the circles a? + y? = 9 and x?…

A: To evaluate the integral.

Q: Use the surface area formula SA = /I. V1 + f% + f; dA to prove that the surface area of the sphere…

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Q: dr Let A be the area of a circle with radius r. If dA when r = 5. dt 5, find dt

A: the area of the circle is given by take the derivative with respect to t ...............(1)

Q: Draw the curve r2 = 2a? $20, a > 0 and calculate the area enclosed by this curve COS

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Q: The volume of the parallelepiped is Volume = |a x b||c|| sin | = |(a × %3D b). c| True False

A: Using direct formula

Q: Find the area of the part of the sphere r? + y + z² = 4 that lies within the first octant and inside…

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Q: Evaluate y°dx - r'dy, where C is the boundary of the region between the circles + y = 1 and 7? + y =…

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Q: Find the area of the upper portion of the cylinder x² + z? = 1 that lies between the planes x = ±1/2…

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Q: Find c > 0 such that the area of the region enclosed by the parabolas y = a? – c² and y = c – a? is…

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Q: Refer to the figure and find the volume V generated by rotating the given region about the specified…

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Q: Use spheri\cal coordinatcs Evaluate y'zdV, where E lies above the cone 4 = /3 and below the Sphere…

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Q: What is the area of the region that lies inside the circle r + y y? = 5x? ² ? %3D 10.r and outside…

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Q: The volume of the parallelepiped is Volume = |a x b||c|| sin o| = |(a x b)- c| %3D %3D True False

A: We have to check either the volume of parallelepiped is volume=a×bcsinϕ=a×b·c or not.

Q: ...) The figure below shows contours of g(x, y) on the region R, with 9 < x < 15 and 2 < y < 8. 7 6…

A: Consider the provided question, R=x,y| 9⩽x⩽15 and 2⩽y⩽8Using ∆x=∆y=2, break the region R into 9…

Q: Find the area of the portion of the sphere x2 + y² + inder x2 + y? = 6y. %3D

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Q: Find the surface area of a sphere of radius r

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Q: B is the first quadrant part of the region enclosed by the circles x2 + y? = a? and x2 + y? = b2…

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Q: Find c > 0 such that the area of the region enclosed by the parabolas y = 390. c and y = c² – a? is…

A: we know that areaA of the region enclosed by given parabola is given by A=2∫-ccc2-x2dx=4∫0cc2-x2dx…

Q: Find the minimum distance from the surface to the given point. (Hint: Start by minimizing the square…

A: The minimizing function is, d2=(x-8)2+y2+z2

Q: Find the surface area of the torus obtained by rotating the circle x2 + (y – b)? =r² about the…

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Q: Refer to the figure and find the volume V generated by rotating the given region about the specified…

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Q: the line integral SexY² ds where C is the of the Circle x² + y²=4 oriented clockwise Evaluate Cight…

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Q: 1) A) Find the area of the portion of the sphere x? + y + =9 lying inside the cylinder x2 + y? = 3y.

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Q: Using Green’s formula, evaluate the line integral ∮C(x−y)dx+ (x+y)dy, where C is the circle…

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Q: Find the area of the part of the cone S = {(x, y, 2); z = Vx² + y?, x²+y² < 1}.

A: Given surface is S=x,y,z:z=x2+y2,x2+y2≤1 We have to find the area of the surface. Here, z=x2+y2…

Q: Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of…

A: We know by the theorem of Pappu that the volume of a solid is obtained by area times the distance…