What is the total area enclosed between the curves $y = x$ and $y = x^{2}$ ?

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Evaluate the definite integral $\int_{0}^{1} \frac{1}{1 + x ^{2}} d x$ .
If $f (x) = \int_{0}^{x} (t - 1) e^{t^{2}} d t$ , where does $f$ attain its local minimum?
Evaluate $\int_{0}^{1} x^{2} e^{x} d x$ using repeated integration by parts.
Evaluate $\int_{0}^{1} \frac{1}{4 - x ^{2}} d x$ .
If continuous function $f$ satisfies $\int_{0}^{x} f (t) d t = x^{2} + 2 x$ for all $x$ , what is $f (3)$ ?
What is the limit of Riemann sums $lim_{n \to \infty} \sum_{i = 1}^{n} \frac{1}{n} \frac{i}{n}$ ?