(1 point)
Evaluate the definite integrals
a) $\int_{0}^{9} \sqrt{t} d t=$
b) $\int_{4}^{25} \frac{1}{\sqrt{z}} d z=$
c) $\int_{1}^{27} \sqrt[3]{x} d x=$

Step 1 ：a) $\int_{0}^{9} \sqrt{t} dt = \frac{2}{3}t^{\frac{3}{2}} + C$

Step 2 ：Substituting the limits of integration: $\int_{0}^{9} \sqrt{t} dt = \frac{2}{3}(9^{\frac{3}{2}}) - \frac{2}{3}(0^{\frac{3}{2}})$

Step 3 ：Simplifying: $\int_{0}^{9} \sqrt{t} dt = \frac{2}{3}(27) - \frac{2}{3}(0) = 18$

Step 4 ：Therefore, $\int_{0}^{9} \sqrt{t} dt = \boxed{18}$

Step 5 ：b) $\int_{4}^{25} \frac{1}{\sqrt{z}} dz = 2\sqrt{z} + C$

Step 6 ：Substituting the limits of integration: $\int_{4}^{25} \frac{1}{\sqrt{z}} dz = 2(\sqrt{25}) - 2(\sqrt{4})$

Step 7 ：Simplifying: $\int_{4}^{25} \frac{1}{\sqrt{z}} dz = 10 - 4 = 6$

Step 8 ：Therefore, $\int_{4}^{25} \frac{1}{\sqrt{z}} dz = \boxed{6}$

Step 9 ：c) $\int_{1}^{27} \sqrt[3]{x} dx = \frac{3}{4}x^{\frac{4}{3}} + C$

Step 10 ：Substituting the limits of integration: $\int_{1}^{27} \sqrt[3]{x} dx = \frac{3}{4}(27^{\frac{4}{3}}) - \frac{3}{4}(1^{\frac{4}{3}})$

Step 11 ：Simplifying: $\int_{1}^{27} \sqrt[3]{x} dx = \frac{3}{4}(81) - \frac{3}{4}(1) = \frac{243}{4} - \frac{3}{4} = \frac{240}{4} = 60$

Step 12 ：Therefore, $\int_{1}^{27} \sqrt[3]{x} dx = \boxed{60}$