Application of Integrals

34 solved questions from Class 12 Mathematics, each with the full working. Tap any question to see the step-by-step solution.

• 1 What is the domain, range, or fundamental theorem introduced in 'Application of Integrals' (Class 12 Mathematics)? medium MCQ Functions & Theorems
• 2 What is the derivative or integral formula associated with 'Application of Integrals'? medium MCQ Calculus & Formulae
• 3 How do you find the solution set or optimum value in 'Application of Integrals'? medium MCQ Problem Solving & Optimization
• 4 A thin rod of length 5 m has a linear mass density λ(x)=2x kg/m, where x is the distance from one end. Find the position of the centre of mass from that end. hard 4 marks Centre of mass of a variable density rod
• 5 Find the volume of the solid obtained by rotating the region bounded by y=√x, x=4, and the x‑axis about the y‑axis using the shell method. easy 2 marks Volume by shell method
• 6 Find the centroid (x,y) of the region bounded by y=√x, the x‑axis and the line x=4. hard 4 marks Centroid of planar region
• 7 Find the area of the region enclosed between the curves y = x and y = x^2 from x = 0 to x = 1. easy 2 marks Area between two curves
• 8 Find the coordinates of the centroid (x,y) of the region bounded by y = sin x, the x‑axis, x = 0 and x = π. hard 5 marks Centroid of a plane region
• 9 A cylindrical tank of radius 5 m is filled with water up to its brim. The water is pumped out through a pipe at the top of the tank. Assuming water density ρ = 1000 kg/m^3 and g = 9.8 m/s^2, find the total work required to empty the tank. (Give your answer in kilojoules, 1 kJ=10^3 J.) medium 3 marks Work done in pumping fluids
• 10 A right circular conical tank has radius 3 m at the top and height 6 m. It is filled with water up to the brim. Find the total work (in kJ) required to pump all the water out through a pipe at the top of the tank, assuming ρ=1000 kg/m^3 and g=9.8 m/s^2. medium 4 marks Work to pump liquid from a tank
• 11 Find the coordinates of the centroid (x,y) of the region bounded by the curve y=ln x, the x‑axis, and the vertical lines x=1 and x=e. hard 5 marks Centroid of plane region
• 12 Using the washer method, find the volume (in m^3) of the solid obtained by rotating the region bounded by y=cos x, y=0, and x=0 to x=π/3 about the x-axis. medium 3 marks Volume by washers
• 13 Find the area of the region enclosed between the curves y=2x and y=x^2 from x=0 to x=2. easy 2 marks Area between curves
• 14 Using the washer method, find the volume (in m^3) of the solid obtained by rotating the region bounded by y=√x, the x‑axis and the vertical lines x=0 and x=9 about the x‑axis. medium 4 marks Volume by washer method
• 15 Find the area of the region enclosed between the curves y=x^3 and y=x from x=0 to x=1. easy 2 marks Area between two curves
• 16 A hemispherical tank of radius 5 m is filled with water. Find the total work (in kJ) required to pump all the water out through a pipe located at the top of the tank. Take ρ =1000 kg/m^3 and g=9.8 m/s^2. hard 5 marks Work done in pumping fluid out of a tank
• 17 A rectangular water tank has length 2 m, breadth 3 m and height 4 m. The tank is full of water and the water is pumped out through a pipe at the top of the tank. Find the total work required to empty the tank. Take water density ρ =1000 kg/m^3 and g=9.8 m/s^2. Give your answer in kilojoules (kJ). easy 2 marks Work in pumping water
• 18 Determine the coordinates of the centroid (x,y) of the region bounded by y=ln x, the x‑axis, x=1 and x=e^2. hard 3 marks Centroid of a plane region
• 19 A region is bounded by y=√x, the x‑axis and the vertical line x=1. Find the volume of the solid generated when this region is revolved about the line x=3 using the method of cylindrical shells. Express your answer in terms of π. hard 4 marks Volume by shell method
• 20 A region is bounded by y=ln x, the x‑axis, x=1 and x=e. Find the volume of the solid generated when this region is revolved about the y‑axis using the method of cylindrical shells. hard 4 marks Volume by shell method
• 21 A cylindrical tank of radius 2 m and height 5 m is filled with water. Water is pumped out through a pipe at the top of the tank. Find the total work required to empty the tank. Take water density ρ =1000 kg/m^3 and g=9.8 m/s^2. Give your answer in kilojoules (kJ). easy 2 marks Work done in pumping fluid
• 22 Find the volume of the solid obtained by rotating the region bounded by y=√x, the x-axis and the vertical lines x=0 and x=4 about the x-axis using the washer (disk) method. medium MCQ 3 marks Volume by washer method
• 23 Determine the coordinates of the centroid (x,y) of the region bounded by y=x and y=x^3 between x=0 and x=1. hard 5 marks Centroid of plane region
• 24 Find the volume (in cubic metres) of the solid generated when the region bounded by y=e^-x, the x‑axis, x=0 and x=1 is revolved about the y‑axis using the cylindrical shell method. medium 3 marks Volume by cylindrical shells
• 25 Determine the coordinates of the centroid (x,y) of the region bounded by y=sin x, the x‑axis, x=0 and x=π. Express your answer in exact form. hard 4 marks Centroid of a plane region
• 26 A right circular conical tank has height 6 m and top radius 2 m. The tank is full of water. Water is pumped out through a pipe at the top of the tank. Find the work required (in kilojoules) to empty the tank. Take ρ=1000 kg/m^3 and g=9.8 m/s^2. Choose the correct value. medium MCQ 3 marks Work to pump liquid from a conical tank
• 27 Find the volume (in cubic metres) of the solid obtained by rotating the region bounded by y = x^2, the x-axis and the vertical lines x=0 and x=2 about the x-axis using the washer (disk) method. easy 2 marks Volume by washer method
• 28 Determine the coordinates of the centroid (x,y) of the region bounded by y=ln x, the x‑axis, x=1 and x=e^2. Give your answer in exact form. hard 5 marks Centroid of a plane region
• 29 Using the method of cylindrical shells, find the volume (in cubic metres) of the solid generated when the region bounded by y=x^2, the x‑axis and the vertical lines x=0 and x=2 is revolved about the line x=3. medium 3 marks Volume by cylindrical shells
• 30 Find the volume (in cubic metres) of the solid obtained by rotating the region bounded by y=√x, the x‑axis and the vertical lines x=0 and x=1 about the x‑axis using the washer (disk) method. easy 2 marks Volume by washer method
• 31 A right circular conical tank has height 4 m and top radius 3 m. The tank is completely filled with water. Water is pumped out through a pipe at the top of the tank. Find the total work required to empty the tank. Take the density of water ρ=1000 kg/m^3 and g=9.8 m/s^2. Choose the correct option. medium MCQ 3 marks Work against gravity (pumping water)
• 32 Compute the volume (in cubic metres) of the solid obtained by rotating the region bounded by y=x, the x‑axis, and the vertical lines x=0 and x=1 about the x‑axis using the washer method. easy 2 marks Volume by washer method
• 33 Determine the coordinates (x,y) of the centroid of the region bounded by y=x^3 and y=x^2 between x=0 and x=1. Express your answer in exact form. hard 4 marks Centroid of region between curves
• 34 Find the volume (in cubic metres) of the solid obtained by rotating the region bounded by y=√x, the x‑axis, and the vertical lines x=0 and x=4 about the horizontal line y=5 using the washer method. hard 5 marks Volume by washer method with shifted axis