Let f(x) be a continuous and positive function and the area bounded by y=f(x), the x -axis and the lines x=0 and x=a is 4a3+sin a sq. units (a 0) . If f((π /2))=k(π )2, then the value of k is

Question

Let f(x) be a continuous and positive function and the area bounded by y=f(x), the x -axis and the lines x=0 and x=a is 4a3+sin a sq. units (a > 0) . If f((π /2))=k(π )2, then the value of k is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Given area is displaystyle ∫ 0a f(x)dx ∴ displaystyle ∫ 0a f(x)dx=4a3+sin a Differentiating w.r.t. a, we get f(a)=12a2+cos a ⇒ f((π /2))=3(π )2 ⇒ k=3

Q. Let $f (x)$ be a continuous and positive function and the area bounded by $y = f (x),$ the $x$ -axis and the lines $x = 0$ and $x = a$ is $4 a^{3} + s ina$ sq. units $(a > 0)$ . If $f (\frac{π}{2}) = k (π)^{2},$ then the value of $k$ is

Given area is $\int_{0}^{a} f (x) d x$
$∴ \int_{0}^{a} f (x) d x = 4 a^{3} + s ina$
Differentiating w.r.t. $a,$ we get
$f (a) = 12 a^{2} + cos a$
$\Rightarrow f (\frac{π}{2}) = 3 (π)^{2}$
$\Rightarrow k = 3$

Q. Let f(x) be a continuous and positive function and the area bounded by y=f(x), the x -axis and the lines x=0 and x=a is 4a3+sina sq. units (a>0) . If f(2π​)=k(π)2, then the value of k is

Given area is ∫0a​f(x)dx ∴∫0a​f(x)dx=4a3+sina Differentiating w.r.t. a, we get f(a)=12a2+cosa ⇒f(2π​)=3(π)2 ⇒k=3

Q. Let $f (x)$ be a continuous and positive function and the area bounded by $y = f (x),$ the $x$ -axis and the lines $x = 0$ and $x = a$ is $4 a^{3} + s ina$ sq. units $(a > 0)$ . If $f (\frac{π}{2}) = k (π)^{2},$ then the value of $k$ is

Given area is $\int_{0}^{a} f (x) d x$
$∴ \int_{0}^{a} f (x) d x = 4 a^{3} + s ina$
Differentiating w.r.t. $a,$ we get
$f (a) = 12 a^{2} + cos a$
$\Rightarrow f (\frac{π}{2}) = 3 (π)^{2}$
$\Rightarrow k = 3$