Упр.365 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1) а = 3, b = 4, f(x) = х2;
2) а = 0, b = 2, f(х) = х3 + 1;
3) а = 1, b = 8, f(х) = корень 3 степени х ;
4) а = 4, и = 9, f(х) = корень х;
5) а = пи/3, b = 2пи/3, f(х) = sinx;
6) а = -пи/6, b = 0, f(х) = cosx.
3 3 6

1. $$S=\int_{3}^{4}x^2\,dx=\left.\frac{x^3}{3}\right|_{3}^{4}=\frac{4^3-3^3}{3}=\frac{64-27}{3}=\frac{37}{3}=12\frac{1}{3}.$$

2. $$S=\int_{0}^{2}(x^3+1)\,dx=\left.\left(\frac{x^4}{4}+x\right)\right|_{0}^{2}=\frac{2^4}{4}+2=4+2=6.$$

3. $$S=\int_{1}^{8}\sqrt[3]{x}\,dx=\int_{1}^{8}x^{1/3}\,dx=\left.\frac{3}{4}x^{4/3}\right|_{1}^{8}=\frac{3}{4}\left(8^{4/3}-1\right).$$

   $$8^{4/3}=(2^3)^{4/3}=2^4=16,$$

   $$S=\frac{3}{4}(16-1)=\frac{45}{4}=11\frac{1}{4}.$$

4. $$S=\int_{4}^{9}\sqrt{x}\,dx=\int_{4}^{9}x^{1/2}\,dx=\left.\frac{2}{3}x^{3/2}\right|_{4}^{9}=\frac{2}{3}\left(9^{3/2}-4^{3/2}\right).$$

   $$9^{3/2}=27,\qquad 4^{3/2}=8,$$

   $$S=\frac{2}{3}(27-8)=\frac{38}{3}=12\frac{2}{3}.$$

5. $$S=\int_{\pi/3}^{2\pi/3}\sin x\,dx=\left.-\cos x\right|_{\pi/3}^{2\pi/3}=-\cos\frac{2\pi}{3}+\cos\frac{\pi}{3}.$$

   $$S=-\left(-\frac{1}{2}\right)+\frac{1}{2}=1.$$

6. $$S=\int_{-\pi/6}^{0}\cos x\,dx=\left.\sin x\right|_{-\pi/6}^{0}=\sin 0-\sin\left(-\frac{\pi}{6}\right).$$

   $$S=0- \left(-\frac{1}{2}\right)=\frac{1}{2}.$$

Ответ

1) $$12\frac{1}{3}$$; 2) $$6$$; 3) $$11\frac{1}{4}$$; 4) $$12\frac{2}{3}$$; 5) $$1$$; 6) $$\frac{1}{2}$$.