Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf

Chapter 7 Application of Definite Integration Ex 7.1 Questions and Answers.

Question 1.
Find the area of the region bounded by the following curves, the X-axis, and the given lines:
(i) y = x⁴, x = 1, x = 5
Solution:


(ii) y = \(\sqrt{6 x+4}\), x = 0, x = 2
Solution:


(iii) \(\sqrt{16-x^{2}}\), x = 0, x = 4
Solution:




(iv) 2y = 5x + 7, x = 2, x = 8
Solution:


(v) 2y + x = 8, x = 2, x = 4
Solution:


(vi) y = x² + 1, x = 0, x = 3
Solution:


(vii) y = 2 – x², x = -1, x = 1
Solution:




Question 2.
Find the area of the region bounded by the parabola y² = 4x and the line x = 3.
Solution:

Required area = area of the region OABO
= 2(area of the region OACO)


Question 3.
Find the area of the circle x² + y² = 25.
Solution:

By the symmetry of the circle, its area is equal to 4 times the area of the region OABO.
Clearly, for this region, the limits of integration are 0 and 5.
From the equation of the circle, y² = 25 – x².
In the first quadrant y > 0
∴ y = \(\sqrt{25-x^{2}}\)
∴ area of the circle = 4(area of region OABO)




Question 4.
Find the area of the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{25}\) = 1
Solution:

By the symmetry of the ellipse, its area is equal to 4 times the area of the region OABO.
Clearly, for this region, the limits of integration are 0 and 2.
From the equation of the ellipse,