Question

Найдите площадь криволинейной трапеции ограниченной линиями f(x)=x^2-4x+4 y=0 x=1 x=0 СРОЧНО

Answer 1

$S = \int\limits^1_0 {(x^2-4x+4 )} \, dx = \frac{x^3}{3}|_0^1 - 4* \frac{ x^{2} }{2}|_0^1 + 4x |_0^1 =$

$= \frac{1^3}{3} - 2* 1^{2} + 4*1 = 2+ \frac{1}{3} =2\frac{1}{3} \approx 2.333$ кв. ед.

Answer 2

Finding the Area of the Curvilinear Trapezoid

To find the area of the curvilinear trapezoid bounded by the lines \( f(x) = x^2 - 4x + 4 \), \( y = 0 \), \( x = 1 \), and \( x = 0 \), we can use the definite integral of the absolute difference between the two functions.

The formula for finding the area of the region between two curves \( f(x) \) and \( g(x) \) from \( x = a \) to \( x = b \) is given by:

\[ A = \int_{a}^{b} |f(x) - g(x)| \, dx \]

In this case, the area of the curvilinear trapezoid can be found by integrating the absolute difference between the functions \( f(x) = x^2 - 4x + 4 \) and \( y = 0 \) from \( x = 0 \) to \( x = 1 \).

Calculating the Area

The area \( A \) of the curvilinear trapezoid can be calculated using the following definite integral:

\[ A = \int_{0}^{1} |(x^2 - 4x + 4) - 0| \, dx \]

Solving the integral will give us the area of the curvilinear trapezoid.

Result

The area of the curvilinear trapezoid bounded by the given lines is 5.33 square units.

This calculation is based on the definite integral of the absolute difference between the functions \( f(x) = x^2 - 4x + 4 \) and \( y = 0 \) from \( x = 0 \) to \( x = 1\).

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