Applications of Integrals
Math ⇒ Calculus
Applications of Integrals starts at 12 and continues till grade 12.
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See sample questions for grade 12
Find the area between y = \cos(x) and y = \sin(x) from x = 0 to x = \pi/2.
Find the area enclosed between y = x and y = x^2 from x = 0 to x = 1.
If a region is bounded by y = x^2 and y = 4, what is the area between the curves from x = -2 to x = 2?
If a tank is being filled at a rate r(t) = 5t liters per minute, how much water is added between t = 0 and t = 4 minutes?
If the acceleration of a car is a(t) = 4t, and its initial velocity is 2 m/s, what is its velocity at t = 3 seconds?
If the cross-sectional area of a solid perpendicular to the x-axis at x is A(x), what is the volume of the solid from x = a to x = b?
If the density of a rod of length L is given by \rho(x) = 2x, what is the total mass of the rod from x = 0 to x = 3?
If the rate of change of a quantity Q(t) is given by \frac{dQ}{dt} = 6t, and Q(0) = 5, what is Q(2)?
If the region bounded by y = x and y = x^3 is rotated about the x-axis from x = 0 to x = 1, what is the volume of the solid formed?
If the region bounded by y = x^2 and y = 2x is rotated about the x-axis from x = 0 to x = 2, what is the volume of the solid formed?
If the region bounded by y = x^2 and y = 2x is rotated about the y-axis, what is the volume of the solid formed from y = 0 to y = 4?
If the region bounded by y = x^2 and y = 4 is rotated about the x-axis, what is the volume of the solid formed from x = -2 to x = 2?
If the region bounded by y = x^2 and y = 4 is rotated about the y-axis, what is the volume of the solid formed from y = 0 to y = 4?
If the velocity of a particle is given by v(t) = 3t^2, what is the displacement from t = 0 to t = 2?
A particle moves along a straight line so that its velocity at time t is v(t) = t^2 - 4t + 3 (in m/s). Find the total distance traveled by the particle from t = 0 to t = 4 seconds.
A region is bounded above by y = \ln(x), below by y = 0, and on the sides by x = 1 and x = e. Find the area of this region.
A solid has a base in the xy-plane bounded by y = x^2 and y = 4. Every cross-section perpendicular to the x-axis is a square. Find the volume of the solid.
Explain how the definite integral can be used to determine the work required to stretch a spring from its natural length to a length L, given that the force required is proportional to the extension (Hooke's Law).
Find the surface area generated by revolving the curve y = x^3 from x = 0 to x = 1 about the x-axis.
If the region bounded by y = x^3 and y = x is rotated about the y-axis from x = 0 to x = 1, what is the volume of the solid formed?
