The relationship between load (\(y\)) in N and elongation (\(x\)) in mm of a cotton fabric is \(y = \sqrt{x}\). If the breaking elongation of the fabric is \(9\) mm, the work of rupture, in N,mm, is _____________.
Calculating Work of Rupture
The work of rupture is defined as the area under the load-elongation curve up to the point of breaking.
Given the relationship between load \(y\) (in N) and elongation \(x\) (in mm) as:
$\(y = \sqrt{x}\)$
The breaking elongation is given as \(x_{max} = 9\) mm.
To find the work of rupture (W), we integrate the load function \(y\) with respect to elongation \(x\) from 0 to the breaking elongation \(x_{max}\):
$\(W = \int_{0}^{x_{max}} y \, dx\)$
Substituting the given values:
$\(W = \int_{0}^{9} \sqrt{x} \, dx = \int_{0}^{9} x^{1/2} \, dx\)$
Now, perform the integration:
$\(W = \left[ \frac{x^{1/2 + 1}}{1/2 + 1} \right]_{0}^{9} = \left[ \frac{x^{3/2}}{3/2} \right]_{0}^{9} = \left[ \frac{2}{3} x^{3/2} \right]_{0}^{9}$
Evaluate the definite integral:
$\(W = \frac{2}{3} (9)^{3/2} - \frac{2}{3} (0)^{3/2}$
$\(W = \frac{2}{3} (9^{1/2})^3 - 0\)$
$\(W = \frac{2}{3} (3)^3\)$
$\(W = \frac{2}{3} (27)\)$
$\(W = 2 \times 9\)$
$\(W = 18\)$
The work of rupture is 18 N.mm.
Important Questions from Application Of Definite Integral (Area)
The equation of a closed curve in two-dimensional polar coordinates is given by $r = \frac{2}{\sqrt{\pi}}(1 - \sin \theta)$. The area enclosed by the curve is ______ (answer in integer).
The area bounded by the curves, $y = \sqrt{x}$, and $y = 8x^2$ is _______________(rounded off to 3 decimal places)
Two straight lines pass through the origin $(x_0, y_0) = (0,0)$. One of them passes through the point $(x_1, y_1) = (1,3)$ and the other passes through the point $(x_2, y_2) = (1,2)$.
What is the area enclosed between the straight lines in the interval $[0, 1]$ on the x-axis?
Consider the equation for a curve, $y = f(x) = x^2 + x$.
The area enclosed by the curve, the x -axis ($y = 0$ line); the vertical lines passing through $x = 1$ and $x = 2$ is _________ (rounded off to 2 decimal places)- The area of the region (rounded off to one decimal place) enclosed between the curves $y = x$ and $y = 3\sqrt{x}$ and between the lines $x = 0$ and $x = 1$ is ________ units.
