Applications of Integrals (Area): JEE Questions & Solutions

The area of the region, inside the ellipse x²+4y²=4 and outside the region bounded by the curves y=|x|-1 and y=1-|x|, is:

If the area of the region {(x, y): 1 - 2x ≤ y ≤ 4 - x², x ≥ 0, y ≥ 0} is α/β, α, β ∈ ℕ, gcd(α, β) = 1, then the value of (α + β) is:

Let the line x = -1 divide the area of the region {(x,y): 1 + x² ≤ y ≤ 3 - x} in the ratio m:n, where gcd(m,n)=1. Then m+n is equal to

The area of the region A = {(x,y): 4x² + y² ≤ 8 and y² ≤ 4x} is

Let the area of the region bounded by the curve y = max { sin x, cos x }, lines x = 0, x = 3π/2, and the x-axis be A. Then, A + A² is equal…

The area of the region enclosed between the circles x² + y² = 4 and x² + (y - 2)² = 4 is:

Let A₁ be the bounded area enclosed by the curves y=x²+2, x+y=8 and y-axis that lies in the first quadrant. Let A₂ be the bounded area…

Let f(α) denote the area of the region in the first quadrant bounded by x=0, x=1, y²=x and y=|α x-5|-|1-α x|+α x². Then (f(0)+f(1)) is…

The area of the region R={(x,y): xy≤ 8, 1≤ y≤ x², x≥ 0} is:

Let P₁: y = 4x² and P₂: y = x² + 27 be two parabolas. If the area of the bounded region enclosed between P₁ and P₂ is six times the area of…

The area of the region, inside the circle (x-2√3)² + y² = 12 and outside the parabola y² = 2√3x is:

The area of the region enclosed by the curves y = x² - 4x + 4 and y² = 16 - 8x is:

If the area of the larger portion bounded between the curves x² + y² = 25 and y = |x - 1| is 1/4(bπ + c), where b, c ∈ ℕ, then b + c is…

If the area of the region {(x, y): -1 ≤ x ≤ 1, 0 ≤ y ≤ a + e^|x| - e^(-x), a > 0} is (e² + 8e + 1)/e, then the value of a is:

The area of the region { (x, y): x² + 4x + 2 ≤ y ≤ |x + 2| } is equal to:

The area of the region enclosed by the curves y = e^x, y = |e^x - 1|, and the y-axis is:

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 bounded by the curves x(1 + y²) = 1 and y² = 2x is:

Let the area of the region {(x, y): 2y ≤ x² + 3, y + |x| ≤ 3, y ≥ |x-1| } be A. Then 6A is equal to:

Let the area enclosed between the curves |y| = 1 - x² and x² + y² = 1 be α. If 9α = βπ + γ; β, γ are integers, then the value of |β - γ|…

If the area of the region {(x, y): |4 - x²| ≤ y ≤ x², y ≤ 4, x ≥ 0} is (80√2/α - β), α, β ∈ ℕ, then α + β is equal to.

The area of the region bounded by the curve y = max{|x|, x|x-2|}, then x-axis and the lines x = -2 and x = 4 is equal to.

The area of the region {(x, y): |x - y| ≤ y ≤ 4√x} is

If the area of the region { (x, y): |x - 5| ≤ y ≤ 4√x } is A, then 3A is equal to.

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:

If the area of the region {(x, y): 1 + x² ≤ y ≤ min(x + 7, 11 - 3x)} is A, then 3A is equal to:

Let the area of the bounded region {(x, y): 0 ≤ 9x ≤ y², y ≥ 3x - 6 } be A. Then 6A is equal to:

Let the area of region {(x,y): x - 2y + 4 ≥ 0, x + 2y² ≥ 0, x+4y² ≤ 8, y ≥ 0} be m/n with m and n coprime. Find m+n.

If the area of the region {(x, y): 0 ≤ y ≤ min(2x, 6x - x²)} is A, then 12A is equal to:

Let the area of the region {(x, y): 0 ≤ x ≤ 3, 0 ≤ y ≤ min(x² + 2, 2x + 2)} be A. Then 12A is equal to:

The area (in square units) of the region bounded by the parabola y² = 4(x - 2) and the line y = 2x - 8 is:

The area of the region {(x,y): y² ≤ 4x, x < 4, xy(x-1)(x-2)/(x-3)(x-4) > 0, x ≠ 3} is:

Let S = (-1, ∞) and f: S → R be defined as: f(x) = ∫₋₁^x (e^t - 1)¹¹ (2t - 1)⁵ (t - 2)⁷ (t - 3)¹² (2t - 10)⁶¹ dt Let p be the sum of…

The area enclosed by the curves xy + 4y = 16 and x + y = 6 is equal to:

The area of the region {(x, y): y² ≤ 2x, y ≥ 4x - 1} is:

If the area of the region bounded by the curves y² - 2y = -x and x + y = 0 is A, then 8A is equal to:

It the area enclosed by the parabolas P₁: 2y = 5x² and P₂: x² - y + 6 = 0 is equal to the area enclosed by P₁ and y = α x, α > 0, then α³…

Let A = {(x, y) ∈ ℝ²: y ≥ 0, 2x ≤ y ≤ √(4 - (x - 1)²)} and B = {(x, y) ∈ ℝ²: 0 ≤ y ≤ min{2x, √(4 - (x - 1)²)}}. Then the ratio of the area…

Let Δ be the area of the region {(x, y) ∈ ℝ²: x² + y² ≤ 21, y² ≤ 4x, x ≥ 1}. Then 1/2(Δ - 21sin⁻¹(2/√7)) is equal to:

The area of the region A = {(x, y): |cos x - sin x| ≤ y ≤ sin x, 0 ≤ x ≤ π/2} is:

Let α be the area of the larger region bounded by the curve y² = 8x and the lines y = x and x = 2, which lies in the first quadrant. Then…

Let q be the maximum integral value of p in [0, 10] for which the roots of the equation x² - px + 5p/4 = 0 are rational. Then the area of…

Let A be the area of the region { (x, y): y ≥ x², y ≥ (1 - x)², y ≤ 2x(1 - x) }.Then 540A is equal to.....

Let the area of the region { (x, y): |2x - 1| ≤ y ≤ |x² - x|, 0 ≤ x ≤ 1 } be A. Then (6A + 11)² is equal to:

Let A be the area bounded by the curve y = x|x - 3| the x-axis, and the ordinates x = -1 and x = 2. Then 12A is equal to:

The area of the region given by {(x, y): xy ≤ 8, 1 ≤ y ≤ x²} is:

If the area of the region S = {(x,y): 2y - y² ≤ x² ≤ 2y, x ≥ y} is equal to (n+2)/(n+1) - π/(n-1), then the natural number n is equal to:

The area bounded by the curves y = |x - 1| + |x - 2| and y = 3 is equal to:

The area of the region {(x, y): x² ≤ y ≤ 8-x², y ≤ 7} is:

If y = p(x) is the parabola passing through points (-1, 0), (0, 1), and (1, 0), and the area of the region {(x, y): (x + 1)² + (y - 1)² ≤…

Area of the region (x, y): x² + (y - 2)² ≤ 4, x² ≥ 2y is:

If A is the area in the first quadrant enclosed by the curve C: 2x² - y + 1 = 0, the tangent to C at the point (1, 3), and the line x + y =…

The area of the region enclosed by the curve y = x³ and its tangent at the point (-1, -1) is:

The area of the region enclosed by the curve f(x) = max {sin x, cos x}, where -π ≤ x ≤ π and the x-axis is:

The area of the region {(x, y): x² ≤ y ≤ |x² - 4|, y ≥ 1} is:

If the area bounded by the curve 2y²=3x, the lines x+y=3, y=0 and lying outside the circle (x-3)²+y²=2 is A, then 4(π+4A) is equal to