Application of Integrals — Class 12 solved problems

28 problems from this chapter, each solved step by step.

Find the area bounded by the curves y = x³ and y = 4x.
The area bounded by the curves y = x³ and y = 4x is 8 square units.
(a) Find the consumer's surplus for the demand function p = 25 - x - x², where the prevailing market price p_0 = 19. OR (b) Solve the following initial value di
The consumer's surplus is (22)/(3).
Find the length of the curve defined parametrically by x = a(cos t + t sin t), y = a(sin t - t cos t) for 0 ≤ t ≤ π/2.
The length of the curve is (aπ²)/(8).
The area of the region, inside the circle (x - 2√3)² + y² = 12 and outside the parabola y² = 2√3 x is:
6π - 16
Find the area of the region bounded by the curves y = x² and y = |x|.
The area of the region bounded by the curves y = x² and y = |x| is (1)/(3) square units.
A cylindrical tank with radius 2 m and height 5 m is filled with water. Find the work required to pump all the water to the top of the tank.
490000π J
Find the area of the region bounded by the curves y² = 4x and x² = 4y.
The area of the region bounded by the curves y² = 4x and x² = 4y is (16)/(3) square units.
The area of the region (x,y): x-y ≤ y ≤ 4√x is:
The area of the region is (1024)/(3).
Find the arc length of the curve y = ln(sec x) from x = 0 to x = π/4.
The arc length of the curve is ln(√2 + 1).
A racing track is build around an elliptical ground whose equation is given by 9x² + 16y² = 144. The width of the track is 3 m as shown below: Based on given in
(i) y = ±(3)/(4)√(16 - x²) (ii) (3)/(4)[(x)/(2)√(16-x²) + 8sin^(-1)((x)/(4))] + C (iii) (a) 12π m² (iii) (b) P=(7, 0), Q=(0, 6) Area of POQ = 21 square units
The area (in sq. units) of the region (x, y): 0 ≤ y ≤ 2|x| + 1, 0 ≤ y ≤ x² + 1, |x| ≤ 3 is
The area of the region is (16)/(3) square units.
If the area of the larger portion bounded between the curves x² + y² = 25 and y = |x - 1| is (1)/(4)(b π + c), b, c N, then b + c is equal to
77
Find the volume of the solid generated by revolving the region bounded by y = x³, y = 0 and x = 2 about the y-axis using the washer method.
The volume of the solid is (64π)/(5) cubic units.
Let the area of the region (x, y): 2y ≤ x² + 3, y + |x| ≤ 3, y ≥ |x-1| be A. Then 6A is equal to:
24
If the area of the region bounded by the curves y = 4 - (x²)/(4) and y = (x - 4)/(2) is equal to α, then 6α equals:
250
EXTENSION: Find the volume of the solid generated by revolving the region bounded by y = x³, y = 0 and x = 2 about the y-axis using the washer method. Additiona
The volume of the solid is (64π)/(5) cubic units.
Let f: R → R be a twice differentiable function such that f(x + y) = f(x) f(y) for all x, y R. If f'(0) = 4a and f satisfies f''(x) - 3a f'(x) - f(x) = 0, a > 0
e² - 1
If the area of the region (x, y): -1 ≤ x ≤ 1, 0 ≤ y ≤ a + e^(|x|) - e^(-x), a > 0 is (π² + 8e + 1)/(e), then the value of a is:
a = (π² - e² + 10e)/(2e)
38. A racing track is build around an elliptical ground whose equation is given by 9x² + 16y² = 144. The width of the track is 3 m as shown below: Based on give
(i) y = ± (3)/(4) √(16 - x²) (ii) (3)/(4) [ (x)/(2) √(16 - x²) + 8 sin^(-1) ((x)/(4)) ] + C (iii) (a) 12π units² (iii) (b) Coordinates P (7,0), Q (0,6), Area 21
MODIFIED: Find the arc length of the curve y = ln(sec x) from x = 0 to x = π/4.
The arc length of the curve is ln(√2 + 1).
Let f: [0, ∞) → R be a differentiable function such that f(x) = 1 - 2x + ∫_0^x e^(t-x) f(t) dt for all x [0, ∞). Then the area of the region bounded by y = f(x)
The area of the region bounded by y = f(x) and the coordinate axes is (5√5-1)/(24).
The area of the region (x, y): x² + 4x + 2 ≤ y ≤ |x + 2| is equal to
(20)/(3)
Let the area enclosed between the curves |y|=1-x² and x²+y²=1 be α. If 9α=βπ+, β, are integers, then the value of |β- | equals:
33
Let the function, f(x) = -3 a x² - 2, & x < 1 a² + b x, & x 1 be differentiable for all x R, where a > 1, b R. If the area of the region enclosed by y = f(x) an
18
If the area of the region (x,y): |x-5| ≤ y ≤ 4√x is A, then 3A is equal to _____
200
The area of the region enclosed by the curves y = e^x, y = |e^x - 1|, and the y -axis is:
1 - ln 2
Let the area of the bounded region (x, y): 0 ≤ 9x ≤ y², y ≥ 3x - 6 be A. Then 6A is equal to _____
81
38. A racing track is build around an elliptical ground whose equation is given by 9x² + 16y² = 144. The width of the track is 3 m as shown below: Based on give
(i) y = ±(3)/(4)√(16 - x²) (ii) (3)/(4)[(x)/(2)√(16 - x²) + 8sin^(-1)((x)/(4))] + C (iii) (a) The area of the region enclosed within the elliptical ground exclu

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