Integration and Accumulation of Change

The product of a series of terms.

The limit of a function.

The sum of a series of terms.

The derivative of a function.

4

2

1

0.5

10

20

30

40

$\sum_{i=1}^{n} \frac{2}{n} \cdot (1 + \frac{2i}{n})^3$

$\sum_{i=0}^{n-1} \frac{2}{n} \cdot (1 + \frac{2i}{n})^3$

$\sum_{i=1}^{n} \frac{3}{n} \cdot (1 + \frac{3i}{n})^3$

$\sum_{i=0}^{n-1} \frac{3}{n} \cdot (1 + \frac{3i}{n})^3$

The height of each subinterval.

The width of each subinterval.

The area under the curve.

The number of subintervals.

$\frac{4i}{n}$

$\frac{4}{n}$

$\frac{i}{n}$

$4 + \frac{i}{n}$

Left Riemann sum because it starts from i=0.

Right Riemann sum because it ends at n-1.

Neither a left nor right Riemann Sum.

Both left and right Riemann sum.

$\int_0^2 x^2 dx$

$\int_1^3 x^2 dx$

$\int_0^2 (1+x)^2 dx$

$\int_1^3 (1+x)^2 dx$

$\lim_{n \to \infty} \sum_{i=1}^{n} \frac{3}{n} \cdot (\frac{3i}{n})^2$

$\lim_{n \to \infty} \sum_{i=0}^{n-1} \frac{3}{n} \cdot (\frac{3i}{n})^2$

$\lim_{n \to \infty} \sum_{i=1}^{n} \frac{3}{n} \cdot i^2$

$\lim_{n \to \infty} \sum_{i=0}^{n-1} \frac{3}{n} \cdot i^2$

$\int_0^2 (x^3 + 1) dx = 6$

$\int_0^2 x^3 dx = 4$

$\int_0^2 (x^3 + 1) dx = 4$

$\int_0^2 x^3 + 1 dx = 6$