hi
\(
a) \\
f(x) = x^4-10x^2+9 \\
Nullstellen\\
x^4-10x^2+9 = 0 \\
z = x^2 \\
z^2-10z +9 = 0 \\
z_{1,2} = 5 \pm \sqrt{25-9} = 5 \pm 4\\
z_1 = 9, z_2 = 1\\
x = \sqrt{z} \\
x_{1,2}= \pm 3\\
x_{2,3}= \pm 1\\
f(x) = x^4-10x^2+9 \\
\int f(x)\mathrm{d}x = \frac{x^5}{5} - 10\frac{x^3}{3} + 9x + C \\
\int_{0}^{1}f(x)\mathrm{d}x = \int_{-1}^{0}f(x)\mathrm{d}x \\
\int_{1}^{3}f(x)\mathrm{d}x = -\int_{-1}^{-3}f(x)\mathrm{d}x \\
A = 2\int_{0}^{1}f(x)\mathrm{d}x + 2\left |\int_{1}^{3}f(x)\mathrm{d}x \right |\\
A = 2 \left [ \frac{x^5}{5} - 10\frac{x^3}{3} + 9x \right ]^1_0 + 2\left |\left [ \frac{x^5}{5} - 10\frac{x^3}{3} + 9x \right ]^3_1 \right | \\
A = 2 \cdot \frac{88}{15}+2\cdot \left | -\frac{304}{15} \right | =
2\cdot \frac{88+304}{15} = \frac{784}{15} \approx 52,267\\
\)
\(
b) \\
f(x) = 0.5x^5 - 2x^3 \\
Nullstellen \\
0.5x^5 - 2x^3 = 0 \\
0.5x^3(x^2-4) = 0 \\
x_1 = 2, x_2 = -2 \\
\int f(x)\mathrm{d}x = 0.5\frac{x^6}{6}-2\frac{x^4}{4} + C = \frac{x^6-6x^4}{12} + C\\
A = 2\left |\int_{0}^{2} f(x)\mathrm{d}x \right |
A = \left |\left [ \frac{x^6-6x^4}{12} \right ]_0^2 \right | \\
A = 2\cdot\left | -\frac{32}{12} \right | = \frac{32}{6} \approx 5.33
\)