integral from -sqrt 5 to -1(-x^2-5)dx+ integral from -1 to 0 (4x) d

To determine the area of the shaded region in the third quadrant more accurately, we need to consider the integrals over the specified intervals. We are given two integrals to calculate:

1. \(\int_{-\sqrt{5}}^{-1} (-x^2 - 5) \, dx\)
2. \(\int_{-1}^{0} (4x) \, dx\)

Let's compute each integral separately and then sum them up.

### Integral 1: \(\int_{-\sqrt{5}}^{-1} (-x^2 - 5) \, dx\)

Evaluate the integral:
\[ \int_{-\sqrt{5}}^{-1} (-x^2 - 5) \, dx = \int_{-\sqrt{5}}^{-1} -x^2 \, dx - \int_{-\sqrt{5}}^{-1} 5 \, dx \]

1. Compute \(\int_{-\sqrt{5}}^{-1} -x^2 \, dx\):
\[ \int_{-\sqrt{5}}^{-1} -x^2 \, dx = -\int_{-\sqrt{5}}^{-1} x^2 \, dx \]
\[ = -\left[ \frac{x^3}{3} \right]_{-\sqrt{5}}^{-1} \]
\[ = -\left( \frac{(-1)^3}{3} - \frac{(-\sqrt{5})^3}{3} \right) \]
\[ = -\left( \frac{-1}{3} - \frac{-5\sqrt{5}}{3} \right) \]
\[ = -\left( -\frac{1}{3} + \frac{5\sqrt{5}}{3} \right) \]
\[ = \frac{1}{3} - \frac{5\sqrt{5}}{3} \]
\[ = \frac{1 - 5\sqrt{5}}{3} \]

2. Compute \(\int_{-\sqrt{5}}^{-1} 5 \, dx\):
\[ \int_{-\sqrt{5}}^{-1} 5 \, dx = 5 \int_{-\sqrt{5}}^{-1} 1 \, dx \]
\[ = 5 \left[ x \right]_{-\sqrt{5}}^{-1} \]
\[ = 5 \left( -1 - (-\sqrt{5}) \right) \]
\[ = 5 (-1 + \sqrt{5}) \]
\[ = 5\sqrt{5} - 5 \]

Summing these results:
\[ \int_{-\sqrt{5}}^{-1} (-x^2 - 5) \, dx = \frac{1 - 5\sqrt{5}}{3} + 5\sqrt{5} - 5 \]
\[ = \frac{1 - 5\sqrt{5} + 15\sqrt{5} - 15}{3} \]
\[ = \frac{1 + 10\sqrt{5} - 15}{3} \]
\[ = \frac{-14 + 10\sqrt{5}}{3} \]

### Integral 2: \(\int_{-1}^{0} (4x) \, dx\)

Evaluate the integral:
\[ \int_{-1}^{0} 4x \, dx = 4 \int_{-1}^{0} x \, dx \]
\[ = 4 \left[ \frac{x^2}{2} \right]_{-1}^{0} \]
\[ = 4 \left( \frac{(0)^2}{2} - \frac{(-1)^2}{2} \right) \]
\[ = 4 \left( 0 - \frac{1}{2} \right) \]
\[ = 4 \left( -\frac{1}{2} \right) \]
\[ = -2 \]

### Total Area

Sum the results of both integrals:
\[ \text{Total Area} = \left( \frac{-14 + 10\sqrt{5}}{3} \right) + (-2) \]
\[ = \frac{-14 + 10\sqrt{5}}{3} - \frac{6}{3} \]
\[ = \frac{-14 + 10\sqrt{5} - 6}{3} \]
\[ = \frac{-20 + 10\sqrt{5}}{3} \]

Therefore, the area of the shaded region is:
\[ \boxed{\frac{-20 + 10\sqrt{5}}{3}} \]

Do you want more details or have any questions?

Here are some related questions you might find helpful:
1. How do you decompose a function into simpler integrals?
2. What are the steps to integrate polynomial functions?
3. How do you interpret definite integrals geometrically?
4. How can you verify the points of intersection between curves?
5. What methods can you use to simplify complex integrals?
6. How does changing the limits of integration affect the integral?
7. What is the significance of negative results in definite integrals?
8. How do you handle integrals with square root limits?

**Tip:** When solving integrals with complex limits, it's useful to break down the integral into smaller, more manageable parts and solve them step by step.

Learn how to calculate the area of a shaded region using definite integrals. This problem involves integrating functions over specified intervals and summing the results. Suitable for advanced high school students and those studying calculus.

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