Question

Вычислить площадь плоской фигуры ограниченной заданными кривыми сделать чертеж. y=1/x y=0 x=1 x=2

Answer 1

y=1/x y=0 x=1 x=2 область на графике.
s=∫1→2  dx/x= - 1/x²=-1/2²+1=3/4

Answer 2

Calculating the Area of the Given Plane Figure

To calculate the area of the plane figure bounded by the given curves, we can use the double integral over the region enclosed by the curves. The given curves are: - $y = \frac{1}{x}$ - $y = 0$ - $x = 1$ - $x = 2$

The area can be calculated using the following double integral: $$\iint_D dA$$ where $D$ is the region enclosed by the curves.

Solution

The region $D$ can be defined by the given curves as follows: - The lower limit of integration for $x$ is 1, and the upper limit is 2. - The lower limit of integration for $y$ is 0, and the upper limit is $\frac{1}{x}$.

Therefore, the double integral to calculate the area is: $$\iint_D dA = \int_{1}^{2} \int_{0}^{\frac{1}{x}} dy dx$$

Let's calculate the area using this double integral.

Calculating the Double Integral

The double integral can be calculated as follows: $$\iint_D dA = \int_{1}^{2} \int_{0}^{\frac{1}{x}} dy dx$$ $$= \int_{1}^{2} \left[y\right]_{0}^{\frac{1}{x}} dx$$ $$= \int_{1}^{2} \left(\frac{1}{x} - 0\right) dx$$ $$= \int_{1}^{2} \frac{1}{x} dx$$

Now, let's calculate the definite integral: $$= \left[\ln|x|\right]_{1}^{2}$$ $$= \ln|2| - \ln|1|$$ $$= \ln(2) - 0$$ $$= \ln(2)$$

Conclusion

The area of the plane figure bounded by the given curves $y = \frac{1}{x}$, $y = 0$, $x = 1$, and $x = 2$ is $\ln(2)$.