Question

Найдите площадь фигуры, ограниченной линиямиy=-x²-1, y=0, x=4, x=1нужно построить параболу

Answer 1

Решение смотрите во вкладке.

Answer 2

Calculation of the Area

To find the area of the figure bounded by the lines y = -x² - 1, y = 0, x = 4, and x = 1, we need to determine the region enclosed by these lines and calculate its area.

The given lines form a closed figure that consists of two parts: a parabolic region and a rectangular region.

1. Parabolic Region: The equation y = -x² - 1 represents a downward-opening parabola. To visualize the parabolic region, we can plot the graph of this equation.


The parabolic region is the area under the curve y = -x² - 1 between x = 1 and x = 4. To find the area of this region, we can integrate the equation y = -x² - 1 with respect to x over the interval [1, 4].

∫[-x² - 1]dx = [-x³/3 - x] from 1 to 4

Evaluating the integral, we get:

[-4³/3 - 4] - [-1³/3 - 1] = [-64/3 - 4] - [-1/3 - 1] = -76/3 + 2/3 = -74/3

Therefore, the area of the parabolic region is 74/3 square units.

2. Rectangular Region: The rectangular region is formed by the lines y = 0, x = 4, and x = 1. It is a rectangle with a base of length 3 (from x = 1 to x = 4) and a height of 0.

Therefore, the area of the rectangular region is 0 square units.

3. Total Area: To find the total area of the figure, we add the areas of the parabolic and rectangular regions:

Total Area = Parabolic Area + Rectangular Area = 74/3 + 0 = 74/3 square units.

Conclusion

The area of the figure bounded by the lines y = -x² - 1, y = 0, x = 4, and x = 1 is 74/3 square units.