A train on a model railway leaves a station and speeds up, and a meter records its speed…

A train on a model railway leaves a station and speeds up, and a meter records its speed. Its speed is $v(t) = t^2$ metres per second for the first $6\ \text{s}$, where $t$ is in seconds. A student wants the distance covered in these $6\ \text{s}$, and she cannot yet use an antiderivative, so she uses rectangles. She cuts the time into $n = 6$ strips of width $1\ \text{s}$. The right-hand sum takes the speed at the end of each strip, and the left-hand sum takes the speed at the start. She knows that the exact distance is the area under the speed graph. She finds the right-hand sum, the left-hand sum, and then the exact distance with an antiderivative. Finally she notes that the exact distance lies between the two sums, because the speed is increasing. She would like to know that the exact value agrees with what the rectangles suggest. Find the right-hand sum, the left-hand sum, and the exact distance, each in metres.