Question

Найти площадь фигуры, ограниченной линиями y=2/x; y=0; x=3; x=1. Сделать чертёж.

Answer 1

Решение смотри на фото.

Answer 2

Calculation of the Area

To find the area of the figure bounded by the lines y = 2/x, y = 0, x = 3, and x = 1, we can break down the figure into two parts: a triangle and a region bounded by a curve.

1. Triangle: - The triangle is formed by the lines y = 0, x = 3, and x = 1. - The base of the triangle is the distance between x = 3 and x = 1, which is 3 - 1 = 2 units. - The height of the triangle is the distance between y = 0 and the curve y = 2/x. - To find the height, we need to find the y-coordinate where the curve intersects the y-axis (when x = 0). - Substituting x = 0 into the equation y = 2/x, we get y = 2/0, which is undefined. - Therefore, the triangle does not have a height and its area is 0.

2. Region bounded by the curve: - The curve y = 2/x bounds the region from x = 1 to x = 3. - To find the area of this region, we need to integrate the curve equation with respect to x over the interval [1, 3]. - The integral of y = 2/x with respect to x is given by ∫(2/x)dx = 2ln|x| + C, where C is the constant of integration. - Evaluating the integral over the interval [1, 3], we get: ∫[1, 3](2/x)dx = [2ln|x|] from 1 to 3 = 2ln|3| - 2ln|1| = 2ln(3) - 0 = 2ln(3).

3. Total Area: - The total area of the figure bounded by the lines y = 2/x, y = 0, x = 3, and x = 1 is the sum of the areas of the triangle and the region bounded by the curve. - Since the area of the triangle is 0, the total area is equal to the area of the region bounded by the curve, which is 2ln(3) square units.

Sketch of the Figure

Here is a sketch of the figure bounded by the lines y = 2/x, y = 0, x = 3, and x = 1:

   |\
   | \
   |  \
   |   \
   |    \
   |     \
   |      \
   |_______\
   1        3

Please note that the sketch is not to scale and is only meant to provide a visual representation of the figure.

I hope this helps! Let me know if you have any further questions.