Download App

Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
Question
Evaluate $\int_{2}^{3} x^{2} d x$
✓

Answer

f(x) is continuous in [2, 3]
$\int_{a}^{b} f(x) d x$ = $\mathop {\lim }\limits_{n \to \infty } h\sum\limits_{r = 0}^{n - 1} f (a + rh)$ where $\mathrm{h}=\frac{\mathrm{b}-\mathrm{a}}{\mathrm{n}}$ 
Here,
$\int_{2}^{3}\left(x^{2}\right) d x$ = $\mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{n}} \right)\sum\limits_{r = 0}^{n - 1} f \left( {2 + \left( {\frac{r}{n}} \right)} \right)$ 
= $\mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{n}} \right)\sum\limits_{r = 0}^{n - 1} {{{\left( {2 + \left( {\frac{r}{n}} \right)} \right)}^2}}$ 
= $\mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{n}} \right)\sum\limits_{r = 0}^{n - 1} {\left( {\frac{{{r^2}}}{{{n^2}}} + 4 + \frac{{4r}}{n}} \right)}$
= $\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\left( {\frac{{(n - 1)(n)(2n - 1)}}{{6{n^2}}} + 4n + \frac{{4(n - 1)(n)}}{{2n}}} \right)$ 
= $\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\left( {\frac{{\left( {{n^2} - n} \right)(2n - 1)}}{{6{n^2}}} + 4n + \frac{{2\left( {{n^2} - n} \right)}}{n}} \right)$ 
= $\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\left(\frac{\left(2 n^{3}-2 n^{2}-n^{2}+n\right)}{6 n^{2}}+4 n+\frac{2\left(n^{2}-n\right)}{n}\right)$ 
= $\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\left( {\frac{{\left( {2{n^3} - 3{n^2} + n} \right) + \left( {24{n^3}} \right) + \left( {12{n^3} - 12{n^2}} \right)}}{{6{n^2}}}} \right)$ 
= $\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\left( {\frac{{38{n^3} - 15{n^2} + n}}{{6{n^2}}}} \right)$ 
= $\mathop {\lim }\limits_{n \to \infty } \left( {\frac{{38{n^3} - 15{n^2} + n}}{{6{n^3}}}} \right)$ 
= $\mathop {\lim }\limits_{n \to \infty } \left( {\frac{{38}}{6}} \right) - \left( {\frac{{15}}{{6n}}} \right) + \left( {\frac{1}{{6{n^2}}}} \right)$ 
= $\frac{38}{6}$ 
= $\frac{19}{3}$ 

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "1 Marks" from Integrals→Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Write the adjoint of the following matrix:
$\left(\begin{array}{c}2&-1\\ 4&3\end{array}\right)$
→
Area lying in the first quadrant and bounded by the circle x2 + y2 = 4 and the lines x = 0 and x = 2 is
→
The binary operation * : R x R $\rightarrow$ R is defined as a * b = 2a + b. Find (2 * 3) * 4.
→
State with reason whether following functions have inverse
f: {1, 2, 3, 4} → {10} with
f = {(1, 10), (2, 10), (3, 10), (4, 10)}
→
In set $A=\{0,1,2,3,4,5\} R$ is equivalence relation where $R =\{(a, b):(a-b)$ is divisible by 2$\}$. Write equivalence class [0].
→
A trust fund has ₹ 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide ₹ 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of: ₹ 2000
→
Find all the points of local maxima and local minima of the function f given by f(x) = 2x3 – 6x2 + 6x +5.
→
Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is a sister of b} is the empty relation and R′ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation.
→
Find $'\lambda'$ when the projection of $\overrightarrow{a}$ = $\lambda$$\hat{\text{i}}+\hat{\text{j}}+\hat{\text{4k}}$ on $\overrightarrow{b}=\hat{\text{2i}}+\hat{\text{6j}}+\hat{\text{3k}}\text{ is 4 units.}$
→
Construct a 2 × 2 matrix, A = $[\text{a}_{\text {ij}}]$, whose elements are given by:

$\text a_{\text {ij}}=\frac{\text i}{\text j} $

→