Find the volume of the solid lying inside x² + y² + z² = 1, lying above the xy-plane and inside z = √√√x² + y² (Spherical co-ordinates may help. This is in the shape of an ice cream cone!).

Q: b) Express the area of the pool, A in terms of I and r. c) Hence, or otherwise show that A = -² +…

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Q: 3. Find the right triangle of largest area inscribed under the parabola y = 4 – r² with its base…

A: Given: y=4-x2 Vertex of the parabola will be at 0,4 and the points where parabola intersect the…

Q: There is an ant on a circular heated plate. The plate temperature is given by f(x, y) = y² – x² + 12…

A: Consider the function fx,y=y2-x2+12, with equality constraint x2+y2=121 (since the radius is given…

Q: b) Evaluate the value of I = S(e³y – 3x)dx + (6xey + y)dy, where C' is the parabola - C (0, 0) to…

A: One has to solve it by Green's Theorem.

Q: Solve the given initial-value problem. y" + 4y = -1, y(π/8) -3/44 4' y(x) = y'(π/8)= = 2

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Q: Write down inequalities to represent (a) the region inside the sphere 2² + y² + 2² = 62 and outside…

A: The given problem is to find the inequalities that represent the given solids of region bounded by…

Q: For the integral x6 - 3x³ — 60x4 +720x²dx, approximate it by using Gaussian quadrature with n = 2,…

A: First we transform the interval 0,1 to -1,1 by  x=b+a2+b-a2u…

Q: Draw a sketch of the cylinder z = -4 - y2.

A: A cartesian coordinate system contains two mutually perpendicular number lines named x-axis and…

Q: 4. Find the surface area of the part of the sphere x² + y² + z² = 1 above the plane z = 1/√2.

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Q: Use calculus to show that the maximum possible area of a triangle with one vertex at (0, 0) and the…

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Q: Find the surface area of the part of the sphere x² + y² + z² = 4 that lies above the cone z = √√x² +…

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Q: 3. Use cylindrical coordinates to evaluate 3 √9-x² [² [³²*² [√²²+³ (2² + y²) ³¹ dzdydz 0 Jo

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Q: ) Find the derivetine of the fnchiog h(x) = I sinhex-7x 4

A: h(x)=74sinh(2x)-7x2Differentiate w.r.t…