Question

If `int_0^3 f(x) dx = 8` then calculate? (A) (i) `int_0^3 2f(x) dx`, (ii) `int_0^3 f(x) + 2 dx` (B) `c` and `d` so that `int_c^d f(x-2) dx`

Answer 1

A) (i) `int_0^3 \ 2f(x) \ dx = 16`
A) (ii) `int_0^3 \ f(x) + 2 \ dx = 14`

B) `c=2`; `d = 5`

Explanation:

We are given that:

`int_0^3 \ f(x) \ dx = 8`

Part (A)

(i) `int_0^3 \ 2f(x) \ dx = 2 \ int_0^3 \ f(x) \ dx`
`" " = 2* 8`
`" " = 16`

(ii) `int_0^3 \ f(x) + 2 \ dx = int_0^3 \ f(x) \ dx + int_0^3 \ 2 \ dx`
`" " = int_0^3 \ f(x) \ dx + 2 \ int_0^3 \ dx`
`" " = 8 + 2[x]_0^3`
`" " = 8 + 2(3-0)`
`" " = 8 + 6`
`" " = 14`

Part (B)

We are given that:

`int_c^d \ f(x-2) \ dx = 8`

The graph of `y=f(x-2)` represents a translation of `y=f(x)` by a shift to the right by two units.

Thus, as we know that `int_0^3 \ f(x) \ dx = 8` then

`c-2 = 0 => c=2`
`d-2 = 3 => d = 5`

Answer 2

`(a)(i): 16; (ii): 14; (b): c=2, d=5.`

Explanation:

Given that, for a fun. `f, int_0^3 f(x)dx=8.`

`(a)(i): int_0^3{2f(x)}dx=2int_0^3f(x)dx=2*8=16.`

`(a)(ii): int_0^3{f(x)+2}dx=int_0^3f(x)dx+int_0^3 2dx,`

`=8+2int_0^3 1dx,`

`=8+2[x]_0^3,`

`=8+2[3-0],`

`=8+6,`

`=14.`

`(b): int_c^df(x-2)dx=8.`

Let, `(x-2)=t," so that, "dx=dt.`

Also, when, `x=c, t=x-2=c-2.`

Similarly, when, `x=d, t=d-2.`

`:. int_c^d f(x-2)dx=int_(c-2)^(d-2)f(t)dt.`

But, `int_c^df(x-2)dx=8=int_0^3 f(x)dx=8=int_0^3f(t)dt.`

`:. int_(c-2)^(d-2)f(t)dt=8=int_0^3f(t)dt.`

Evidently, `c-2=0 rArr c=2, and, d-2=3 rArr d=5.`

`;. (c,d)=(2,5).`

Enjoy Maths.!