68 free MCQs on Application of Integrals, each with its own worked answer and explanation. Using definite integrals to compute the area under a curve, the area between two curves, and the areas enclosed by standard curves like circles, parabolas, and ellipses.
Take the timed Application of Integrals chapterwise test →68 practice questions on Application of Integrals, sorted Easy → Hard. Try each one first, then open its answer page for the worked explanation. Want the full theory first? Read the Application of Integrals notes.
The definite integral ∫ₐᵇf(x)dx computes the exact area of the shaded region bounded by the curve, the x-axis, and the vertical lines x=a and x=b - the same idea behind the Riemann sum, but evaluated exactly rather than approximated by rectangles.
Q1.
The area under the curve y = f(x), above the x-axis, between x = a and x = b is given by:
Q2.
If the curve lies entirely below the x-axis over [a,b], the area is:
Q3.
The area between two curves f(x) and g(x) (with f(x) >= g(x)) from x=a to x=b is:
Q4.
The area of a circle x<sup>2</sup> + y<sup>2</sup> = a<sup>2</sup> found using integration is:
Q5.
The area enclosed by the ellipse x<sup>2</sup>/a<sup>2</sup> + y<sup>2</sup>/b<sup>2</sup> = 1 is:
Q7.
Evaluate the area under y = 3 (a constant function) from x = 1 to x = 5.
Q8.
If a curve crosses the x-axis within [a,b], the correct method to find total area is to:
Q9.
Area bounded by curve x = g(y), the y-axis, and lines y=c, y=d is given by:
Q10.
Find the area under y = x<sup>2</sup> from x = 0 to x = 3.
Q11.
The first step in solving an area-between-curves problem should be to:
Q13.
When set up correctly, area is always expressed as a:
Q14.
The area enclosed by a semicircle of radius r above the x-axis (y = sqrt(r<sup>2</sup>-x<sup>2</sup>)) is:
Q15.
The definite integral of a function between two limits gives the ___ under the curve:
Q16.
Integration is the reverse process of:
Q17.
The area under a curve y = f(x) from x = a to x = b is given by:
Q18.
Physical area is always taken to be a ___ quantity:
Q19.
The symbol ∫ represents:
Q20.
In a definite integral, the numbers a and b are called the ___ of integration:
Q21.
Find the area enclosed between y = x and y = x<sup>2</sup> for x in [0,1].
Q22.
Find the area bounded by y = x<sup>2</sup> and y = 4 (the horizontal line), between their intersection points.
Q24.
Find the area under y = cos x from x = 0 to x = pi (careful with sign change at pi/2).
Q25.
Find the area of the region bounded by y<sup>2</sup> = 4x and the line x = 4 (right half of the parabola's enclosed area, upper part only).
Q26.
Find the area between the curves y = x<sup>3</sup> and y = x for x in [0,1].
Q27.
Find the area enclosed by the circle x<sup>2</sup> + y<sup>2</sup> = 9 in the first quadrant only.
Q28.
Find the area bounded by y = x<sup>2</sup> - 4 and the x-axis between x = -2 and x = 2.
Q29.
Find the area of the region in the first quadrant bounded by y = 4 - x<sup>2</sup>, the x-axis, and the y-axis.
Q30.
Find the area enclosed between the line y = x + 2 and the parabola y = x<sup>2.</sup>
Q31.
Find the area bounded by y = 1/x, x = 1, x = e, and the x-axis.
Q32.
Find the area between the curve y = x<sup>2</sup> + 1 and the x-axis from x = 0 to x = 2.
Q33.
The area bounded by y = x², the x-axis and x = 0 to x = 3 is ∫₀³ x² dx =:
Q34.
Found by integration, the area of a full circle of radius r is:
Q35.
The area between an upper curve f(x) and a lower curve g(x) is ∫[f(x) − g(x)] dx taken over their:
Q37.
If a curve lies below the x-axis, its definite integral gives a ___ value:
Q40.
To find an area with respect to the y-axis, we integrate x with respect to:
Q41.
Find the area of the region bounded by the parabola y<sup>2</sup> = 4ax and its latus rectum x = a (full region, both above and below the x-axis).
Q42.
Find the area common to the circle x<sup>2</sup>+y<sup>2</sup> = 4 and the line x = 1 (area of the circular region to the right of x=1).
Q43.
Find the area bounded by y = |x - 1| and the x-axis between x = 0 and x = 3.
Q44.
Find the area enclosed between the parabola y = x<sup>2</sup> and the line y = 4x.
Q45.
Find the area of the smaller region bounded by the ellipse x<sup>2</sup>/9 + y<sup>2</sup>/4 = 1 and the line x/3 + y/2 = 1.
Q46.
Find the area bounded by the curves y = x<sup>2</sup> and y = |x|.
Q47.
Find the area bounded by y = sin x and y = cos x between x = 0 and x = pi/2.
Q48.
Find the area of the region bounded by y = x<sup>3</sup> - x and the x-axis between x = -1 and x = 1.
Q49.
Find the area bounded by the curve y = 4 - x<sup>2</sup> and the lines x = -1 and x = 2 (curve stays above axis throughout).
Q50.
Find the area bounded by the curve y = sqrt(x), the x-axis, and the lines x = 0 and x = 4.
Q51.
The area of the region bounded by the parabola y² = 4ax and its latus rectum is:
Q52.
The area enclosed by the ellipse x²/a² + y²/b² = 1 is:
Q54.
The area enclosed between y = x and y = x² from x = 0 to x = 1 is:
Q56.
The area of the circle x² + y² = a², obtained by integration, equals:
Q57.
The total area between y = x³ and the x-axis from x = −1 to x = 1 (adding absolute areas) is:
Q59.
The area under a curve, being a two-dimensional measure, has units of:
Q60.
The area enclosed by one arch of y = sin x (from 0 to π) is:
Q61.
The area bounded by y = x², the x-axis and the lines x = 0 and x = 3 is:
Q62.
The area enclosed between the curves y = x² and y = x is:
Q63.
The area enclosed between y² = 4x and x² = 4y is:
Q64.
The area enclosed by the ellipse x²/9 + y²/4 = 1 is:
Q66.
The area bounded by y = x³, the x-axis and the lines x = 0 and x = 2 is:
Q67.
The area between y = √x and y = x from x = 0 to x = 1 is:
Q68.
The area of the part of the disc x² + y² = 16 lying in the first quadrant is:
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