Application of Integrals Test

Result

Application of Integrals Test - 42

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Weekly Quiz Competition

Question 1
1 / -0

The area of the region bounded by the curves $$ 1-y^{2}= \left | x \right | and \left | x \right |+\left | y \right |= 1 $$ is

A
$$\frac{1}{3}sq. unit $$

B
$$\frac{2}{3}sq. unit $$

C
$$\frac{4}{3}sq. unit $$

D
$$1 sq.unit$$
Question 2
1 / -0

Find the area of the region enclosed by the curves $$y=x\ \log x$$ and $$y=2x-2x^{2}$$.

A
$$1/12$$

B
$$1/4$$

C
$$2/12$$

D
$$7/12$$

Solution
Question 3
1 / -0

Area of the region bounded by $$x^{2}+y^{2}-6y\leq 0$$ and $$3y\leq x^{2}$$ is

A
$$\frac{9\pi }{2}-12$$

B
$$\frac{9\pi }{4}-6$$

C
$$9\pi$$-24

D
$$\frac{9\pi }{2}+6$$

Solution
Question 4
1 / -0

The area enclosed by the curves y = cosx - sin x and y = [socx - sin x] and between x = 0 and $$x =\dfrac{\pi}{2}$$ is

A
$$2(\sqrt{2} + 1)$$ sq. units

B
$$2(\sqrt{2} - 1)$$ sq. units

C
$$(\sqrt{2} - 1)$$ sq. units

D
$$(\sqrt{2} + 1)$$ sq. units
Question 5
1 / -0

The area (in sq. units) bounded by the parabola $$y = x^{2} - 1$$, the tangent at the point $$(2, 3)$$ to it and the y-axis is

A
$$\dfrac {14}{3}$$

B
$$\dfrac {56}{3}$$

C
$$\dfrac {8}{3}$$

D
$$\dfrac {32}{3}$$

Solution

Here, $$y = x^{2} - 1$$

So, Equation of tangent at $$(2, 3)$$ on
$$\Rightarrow$$ $$y =x^{2} - 1$$, is $$y = (4x - 5) ......(i)$$

$$\therefore$$ Required shaded area
$$\displaystyle = area (\triangle ABC) - \int_{-1}^{3} \sqrt {y + 1} dy$$

$$= \dfrac {1}{2} \cdot (8)\cdot (2) - \dfrac {2}{3} \left ((y + 1)^{3/2}\right )_{-1}^{3}$$

$$= 8 - \dfrac {16}{3} = \dfrac {8}{3}$$ (square units).
Question 6
1 / -0

The area bounded by the parabola $${{\text{y}}^{\text{2}}}{\text{ = 4}}\;{\text{ax}}\;{\text{and}}\;{{\text{x}}^{\text{2}}}\;{\text{ = }}\;{\text{4ay}}\;$$ is

A
$$\dfrac{{8{a^2}}}{3}$$

B
$$\dfrac{{16{a^2}}}{3}$$

C
$$\dfrac{{32{a^2}}}{3}$$

D
$$\dfrac{{64{a^2}}}{3}$$

Solution
Question 7
1 / -0

The area ehclosed by the curves y = f(x) and y =g(x), where f9x) = max $${x , x^2}$$ and g(x) = min $${x, x^2}$$ opver the interval [0,1] is

A
$$\dfrac{1}{6}$$

B
$$\dfrac{1}{3}$$

C
$$\dfrac{1}{2}$$

D
1
Question 8
1 / -0

The area of the region bounded by the parabolas $$y^2= and x^2 = y, is$$

A
$$\dfrac{1}{3}$$ q.units

B
$$\dfrac{8}{3}$$ q.units

C
$$\dfrac{16}{3}$$ q.units

D
$$\dfrac{4}{3}$$ q.units

Solution
Question 9
1 / -0

If the area of the region bounded by the curves, $$y=x^{2},y=\frac{1}{x}$$ and the lines y=0 and x=t (t > 1) is 1 sq. unit, then t is equal to:

A
$${\dfrac{10}{3}}$$

B
$$\dfrac{4}{3}$$

C
$$\dfrac{7}{3}$$

D
$$\dfrac{11}{3}$$

Solution
Question 10
1 / -0

The region in the $$xy$$ - plane is bounded by curve $$y=\sqrt {(25-x^2)}$$ and the line $$y=0$$. If the point $$ (a,a+1)$$ lies in the interior of the region, then

A
$$a \in \left( { - 4,3} \right)$$

B
$$a \in (- \infty, -1) \cup (3, \infty)$$

C
$$a \in (-1,3) $$

D
None of these

Solution