Define the function by s(t)= cases t^ 2 -4t, & for 0 t 4, ! ( t 2 ), & for t>4

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

Define the function

s

s(t)=\begin{cases} t^{2}-4t, & \text{for } 0\le t\le 4,\\ \sin\!\left(\dfrac{\pi t}{2}\right), & \text{for } t>4. \end{cases}

Let

S

be an antiderivative of

s

such that

S(2)=3

. (a) For

0\le x\le 4

, derive an explicit expression for

S(x)

in terms of

x

. (b) Determine the exact value of

S(4)

. (c) For

x\ge 4

, write an expression for

S(x)

that is valid on the interval

4\le x\le 6

. (d) Compute the exact value of

S(6)

. Show all steps leading to your answer.