Following code can be used to retrieve collection of ‘rolled up’ activities for couple of standard entities (account, contact and opportunity). You can find detailed description here.

var target = {
    "@odata.id": "accounts(F591DEE6-5B2F-E711-8110-C4346BAC5238)"
};

var query = {
    "@odata.type": "Microsoft.Dynamics.CRM.QueryExpression",
    "EntityName": "activitypointer",
    "ColumnSet": {"AllColumns":"true"}
};

var requestUrl = "/api/data/v8.2/Rollup(Target=@p1,RollupType=@p2,Query=@p3)";
requestUrl += "?@p1=" + JSON.stringify(target);
requestUrl += "&@p2=Microsoft.Dynamics.CRM.RollupType'Extended'";
requestUrl += "&@p3=" + JSON.stringify(query);

var context;

if (typeof GetGlobalContext === "function") {
    context = GetGlobalContext();
} else {
    context = Xrm.Page.context;
}
var req = new XMLHttpRequest();
req.open("GET", context.getClientUrl() + requestUrl, true);
req.setRequestHeader("OData-MaxVersion", "4.0");
req.setRequestHeader("OData-Version", "4.0");
req.setRequestHeader("Accept", "application/json");
req.setRequestHeader("Content-Type", "application/json; charset=utf-8");
req.onreadystatechange = function () {
    if (this.readyState === 4) {
        req.onreadystatechange = null;
        if (this.status === 200) {
            var results = JSON.parse(this.response);
            //you've got results to results variable
        } else {
            var errorText = this.responseText;
            //handle error here
        }
    }
};
req.send();