Question:
Let [.] denote the greatest integer function then the value of $\int\limits^{1.5}_{0}x \left[x^{2}\right]$ dx is : .
Let [.] denote the greatest integer function then the value of $\int\limits^{1.5}_{0}x \left[x^{2}\right]$ dx is : .
Updated On: Jul 5, 2022
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- $\frac{3}{2}$
- $\frac{3}{4}$
- $\frac{5}{4}$
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The Correct Option is C
Solution and Explanation
$\int\limits^{1}_{0} x\left[x^{2}\right]dx+\int\limits^{\sqrt{2}}_{1}x\left[x^{2}\right]dx+\int\limits^{1.5}_{\sqrt{2}}x\left[x^{2}\right]dx$
$\int\limits^{1}_{0}x.0dx +\int\limits^{\sqrt{2}}_{1}xdx +\int\limits^{1.5}_{\sqrt{2}}2x \,dx$
$0+\left[\frac{x^{2}}{2}\right]^{\sqrt{2}}_{1}+\left[x^{2}\right]^{1.5}_{\sqrt{2}}$
$\frac{1}{2}\left(2-1\right)+\left(2.25-2\right)$
$\frac{1}{2}+.25$
$\frac{1}{2}+\frac{1}{4} = \frac{3}{4}$
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Concepts Used:
Integral
The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.
- The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
- The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
- An integral of a function over an interval on which the integral is described.
Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.
F'(x) = f(x)
For every value of x = I.
Types of Integrals:
Integral calculus helps to resolve two major types of problems:
- The problem of getting a function if its derivative is given.
- The problem of getting the area bounded by the graph of a function under given situations.