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