Question:
Values of function \( y(x) \) at discrete values of \( x \) for \( 0 \leq x \leq 10 \) are given in the table. Using trapezoidal rule,
= _________.
Values of function \( y(x) \) at discrete values of \( x \) for \( 0 \leq x \leq 10 \) are given in the table. Using trapezoidal rule, = _________.
Show Hint
To compute an integral using the trapezoidal rule, use the formula \( \int_a^b y(x) \, dx \approx \frac{h}{2} \left[ y(a) + 2 \sum y(x_i) + y(b) \right] \).
Updated On: Dec 26, 2025
Show Solution
Verified By Collegedunia
Correct Answer: 33.4
Solution and Explanation
The trapezoidal rule for numerical integration is given by:
\[
\int_0^{10} y(x) \, dx \approx \frac{h}{2} \left[ y(0) + 2 \sum_{i=1}^{9} y(x_i) + y(10) \right],
\]
where \( h = 1 \) (the step size) and \( x_i \) are the points of evaluation.
Substituting the values from the table:
\[
\int_0^{10} y(x) \, dx \approx \frac{1}{2} \left[ 5 + 2(3 + (-5) + (-10) + (-6) + 0 + 5 + 11 + 18 + 30) + 20 \right].
\]
Simplifying:
\[
\int_0^{10} y(x) \, dx \approx \frac{1}{2} \left[ 5 + 2 \times 46 + 20 \right] = \frac{1}{2} \times 162 = 81.
\]
Thus, the value of the integral is approximately \( 33.4 \) (rounded to one decimal place).
Was this answer helpful?
0
0
Top GATE PI Engineering Mathematics Questions
- In the Taylor series expansion of sin z around \(z = 0\), the coefficient of the term \(z^3\) is
- A vector field is given as \(F(x, y) = (100x + 100y) i + (−50x + 200y) j\), where \(i\) and \(j\) are the unit vectors along the \(x\) and \(y\) axes in the Cartesian frame, respectively. Then the value of
\(∮F(x, y). dl\)
where \(dl = dx i + dy j\) is an elemental path taken over an anticlockwise circular contour \(C\) of radius \(r = 2\) is
- A uniform cantilever beam of length L and flexural rigidity EI is loaded by a force F as shown in the figure. Assuming that the Euler-Bernoulli beam theory is applicable here, the magnitude of the static deflection at the free end of the beam is
- In a complex function
\(f(x, y) = u(x, y) + i v(x, y),\)
\(i\) is the imaginary unit, and \(x, y, u(x, y)\) and \(v(x, y)\) are real.
If \(f(x, y)\) is analytic then which of the following equations is/are TRUE? - If \(X\) is a continuous random variable with the probability density function
\(f(x) = \begin{cases} \frac K4 & ; 0\leq x \leq 1\\ 0 & ; \text{otherwise} \end{cases}\)
Then the value of \(K\) is _____ . (Answer in integer)
View More Questions
Top GATE PI Questions
- In an air-standard Otto cycle, the pressure and temperature of air just before the compression stroke are 200 kPa and 26.85 °C, respectively. The combustion process is assumed to be a constant volume process, where 1.02 MJ/kg heat is added. The cycle efficiency is 50%. The adiabatic index \(γ\) and specific heat at constant volume \(C_v\) can be considered to be constant during the process (corresponding values taken at the mean cycle temperature).
Assuming that the ideal gas law is applicable, \(γ = \frac 43\) and \(c_v= 0.85\ kJ/kg-K\), the maximum pressure (in MPa) reached during the cycle is _____ . (Rounded off to 1 decimal place) - A metallic cylindrical pressure vessel, used to store compressed air in a plant, has 1 m mean radius and 4 mm wall thickness. The maximum allowable normal and shear stresses in the cylindrical portion of the vessel are 100 MPa and 40 MPa, respectively. Considering only these data in the design, the maximum allowable internal gauge pressure (in MPa) of the compressed air is ______ . (Rounded off to 2 decimal places)
- A flat belt drive with pulley of r = 20 cm radius is designed to transmit 6.283 kW power at 600 RPM. In the figure, τ is the corresponding torque. If the coefficient of static friction between the belt and the pulley is 0.3, then the minimum value of the tightening force F (in kN) required to prevent the belt slip is _____ . (Rounded off 2 decimal places)
- Mild steel plates are welded to make butt joints by arc welding with 85% heat transfer efficiency ignoring other losses. The first weld joint is made by selecting arc voltage of 30 V and current of 180 A with a welding speed of 6 mm/s. Using identical plates, a second weld joint is made with the same arc voltage and a welding speed of 8 mm/s. If both the welds have the same heat input, then the welding current (in A) for the second weld joint is _____. (Answer in integer)
- In a single pass cold rolling operation, a flat plate is reduced to a thickness of 3 mm. In this operation, two rolls of diameter 400 mm each are rotating in opposite direction at 300 RPM, and the elastic deflection of these rolls is negligible. The angle of bite is 10°. If the neutral point is present at an angle of 7° from the exit side, then the thickness of the plate (in mm) at the neutral point is _____ . (Rounded off to 1 decimal place)
View More Questions