5.2: The Definite Integral
- Page ID
- 144293
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- Express \(\displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} (x_i^*) \Delta x\) over \([1, 3]\) as a definite integral.
- Express \(\displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} \sin^2(2\pi x_i^*) \Delta x\) over \([0, 1]\) as a definite integral.
- Evaluate \(\displaystyle \int_{-2}^4 5\ dx\) using formulas from geometry.
- Evaluate \(\displaystyle \int_1^3 (3 - x)\ dx\) using formulas from geometry.
- Evaluate \(\displaystyle \int_{-2}^1 (2x + 1)\ dx\) using formulas from geometry.
- Evaluate \(\displaystyle \int_{-1}^2 (2 - |x|)\ dx\) using formulas from geometry.
- Evaluate \(\displaystyle \int_{-3}^3 \sqrt{9 - x^2}\ dx\) using formulas from geometry.
- Given that \(\displaystyle \int_0^1 x\ dx = \dfrac{1}{2}\), \(\displaystyle \int_0^1 x^2\ dx = \dfrac{1}{3}\), and \(\displaystyle \int_0^1 x^3\ dx = \dfrac{1}{4}\), evaluate \(\displaystyle \int_0^1 (1 - 2x - 3x^2 + 8x^3)\ dx\).
- Given that \(\displaystyle \int_0^1 x\ dx = \dfrac{1}{2}\), \(\displaystyle \int_0^1 x^2\ dx = \dfrac{1}{3}\), and \(\displaystyle \int_0^1 x^3\ dx = \dfrac{1}{4}\), evaluate \(\displaystyle \int_0^1 (1 - 2x)^3\ dx\).
- Use a Riemann sum with the formulas \(\displaystyle \sum_{i=1}^n 1 = n\) and \(\displaystyle \sum_{i=1}^n i = \dfrac{n(n+1)}{2}\) to evaluate \(\displaystyle \int_0^4 (2x - 5)\ dx\).
- Use a Riemann sum with the formulas \(\displaystyle \sum_{i=1}^n i = \dfrac{n(n+1)}{2}\) and \(\displaystyle \sum_{i=1}^n i^2 = \dfrac{n(n+1)(2n+1)}{6}\) to evaluate \(\displaystyle \int_{-2}^{2} (3x^2 - 2x)\ dx\).