Question

❤☺✌☺❤☺✌☺
This Question is only for BRAINLY CHALLENGERS & Maths Aryabhatta

⚠⚠⚠⚠Spammer stay away​

Answer 1

We're given,

$\displaystyle\int\limits_n^{n+1}f(x)\ dx=n^2$

But we have to find,

$\displaystyle\int\limits_{-2}^{4}f(x)\ dx$

In the integral what we're given, the lower limit and the upper limit are two consecutive integers, n and n + 1.

But in the integral what we have to find, the lower limit and the upper limit are respectively -2 and 4, which are not consecutive integers.

So we want to make them as consecutive.

We make use of integration to find the area between the graph and x - axis. Definite integrals are used for finding the area under the graph having a particular domain (inclusively between the upper limit and the lower limit).

Consider the integral what we're given.

$\displaystyle\int\limits_n^{n+1}f(x)\ dx=n^2$

This integral means that the area between the graph of f(x) and x - axis having domain [n, n+1] is n².

And also, consider the integral what we have to find.

$\displaystyle\int\limits_{-2}^{4}f(x)\ dx$

This means we have to find the area between the graph of f(x) and the x - axis having domain [-2, 4].

Consider the domain interval, [-2, 4]. First of all, we have to break this domain as those, each are inclusively between two consecutive integers!

We break [-2, 4] as,

$[-2,\ 4]\ =\ [-2,\ -1] \cup [-1,\ 0] \cup [0,\ 1] \cup [1,\ 2] \cup [2,\ 3] \cup [3,\ 4]$

Hence the integral becomes,

$\displaystyle\int\limits_{-2}^{4}\!f(x)dx\ =\ \int\limits_{-2}^{-1}\!f(x)dx+\int\limits_{-1}^{0}\!f(x)dx+\int\limits_{0}^{1}\!f(x)dx+\int\limits_{1}^{2}\!f(x)dx+\int\limits_{2}^{3}\!f(x)dx+\int\limits_{3}^{4}\!f(x)dx$

And we have,

$\displaystyle\int\limits_n^{n+1}f(x)\ dx=n^2$

So,

$\displaystyle\int\limits_{-2}^{-1}\!f(x)dx=(-2)^2=4\\ \\ \\ \int\limits_{-1}^{0}\!f(x)dx=(-1)^2=1\\ \\ \\ \int\limits_{0}^{1}\!f(x)dx=0^2=0\\ \\ \\ \int\limits_{1}^{2}\!f(x)dx=1^2=1\\ \\ \\ \int\limits_{2}^{3}\!f(x)dx=2^2=4\\ \\ \\ \int\limits_{3}^{4}\!f(x)dx=3^2=9$

$\displaystyle\int\limits_{-2}^{4}\!f(x)dx=4+1+0+1+4+9=\mathbf{19}$

Hence the answer is (C) 19.

Answer 2

Heya........

Answer : (C) 19 ☑️

Plz Refer to the attachment for required answer....... ✔️✔️✔️

✨✨✨❤Hope it helps❤✨✨✨